This document discusses Euler's identity, which is the equation e^iπ + 1 = 0. It provides background on Leonhard Euler, who discovered the identity. The identity relates five important mathematical constants: e, i, π, 0, and 1. Applications of Euler's identity include representing complex numbers in polar form and solving differential equations. In electrical engineering, it underlies functions of capacitance and reactance and is used with signals involving trigonometric and complex exponential functions.

2. Group Members
✘Hamza Farooq 14-EE(E&T)-051
✘M Ahmad Kamal 14-EE(E&T)-045
✘Arslan Riaz 14-EE(E&T)-047

3. Euler’s identity

4. Table OF Contents
✘Introduction of Euler’s identity.
✘What is Euler’s identity?
✘Why Euler’s identity required?
✘Practical applications.
✘Uses in Electrical Engineering.
✘Conclusion
✘References

5. Introduction
✘ Leonhard Euler(1710-1783).
✘Wrote a total of 886 works.
✘The epitome of his mathematical analysis is
summed up in his formula.

6. ✘ According to 1800s mathematician Benjamin
Peirce:
“It is absolutely paradoxical; we cannot
understand it, and we don't know what it means,
but we have proved it, and therefore we know it
must be the truth”.

7. What is Euler’s identity
✘ e is the base of natural logarithms.
✘ iota is the imaginary unit, which satisfies i2 = −1,
✘ pi the ratio of the circumference of a circle to its diameter .
✘The number0, the additive identity.
✘The number1, the multiplicative identity
✘Euler's Identity is also sometimes called Euler's Equation.

8. Continue.
✘ The identity is a special case of Euler's
formula from complex analysis, which states
that
for any real number x. In particular
Since
which gives the identity.

9. Why We need Euler identity?
✘ . iota is the imaginary unit, which satisfies i2 = −1
✘e is the base of natural logarithms.
✘ e^i (imaginary power of exponent) .

10. Applications
✘ Euler’s formula represents polar form of a
complex number.
✘Euler’s formula provides us a convenient way to
move in a circle.
✘Well in differential equations, which may be
differential equations solving phenomena that we
see in nature (such as the weather), you may find
that you need Euler's identity to simplify systems
that have complex solutions.

11. Continue
✘It is used in so many contexts that it is like
asking for the applications of matrix
multiplication.
✘If you “don't understand why it is useful”, try
picturing electrical engineering without it. Or any
other branch of science, which studies
((sinusoidal)) signals of any kind ((acoustic, visual,
magnetic, etc)

12. Continue:
✘The Sylvester-Gallai Theorem.
✘Pick’s Theorem.
.

13. Uses in Electrical engineering
✘ Euler’s formula abounds in electronics and
engineering. Underlying electric functions and
laws of capacitance and reactance is the famous
identity.
✘It is implemented in linear, time-invariant
function input-output machines, otherwise
known as LTI boxes.
✘The particular use of Euler is in electrical
engineering is in case of .

14. Continued:
✘The trigonometric functions and e raised to
powers involving imaginary numbers is a form of
Euler’s Identity that electrical engineers use
regularly.
✘ The context of this equation is bandwidths
mainly in radio station wavelengths.

15. Continued:
✘Ideally, engineers could put a radio signal
through an LTI (Linear time-invariant) box and
get a “zero phase distortion” meaning that a
signal is perfectly unaltered as it passes through
the filter. And for obvious reasons, the less a
sound wave is distorted as it is broadcast, the
better is the quality.

16. Refrences:
✘http://betterexplained.com/articles/intuitive-understanding-of-
eulers-formula/
✘http://www.livescience.com/51399-eulers-identity.html
✘http://www.mathscareers.org.uk/article/beautiful-equation/
✘http://tetrahedral.blogspot.com/2011/01/our-jewel-eulers-
formula-and-eulers.html
✘https://www.researchgate.net/post/What_is_Special_in_Eulers_id
entity_e_ipi_102

17. Credits
Special thanks to:
✘ Presentation template by Ahmad Kamal
✘ Photographs by Ahmad Kamal

18. THANKS!
Any questions?