Pi is the ratio of a circle's circumference to its diameter. It has taken hundreds of algorithms to estimate pi's value as an irrational number. Pi has been developed alongside mathematics and is used in many formulas and applications across engineering, construction, and other fields. It is celebrated on March 14th due to its approximate value of 3.14.

2. GROUP MEMBERS

3. Objectives:
 Study the history of pi.
 Learn the meaning of pi.
 Examine the development of pi along with
the development of mathematics.
 Evaluate some of the different formulas
that pi is used in.
 Explore professions that use pi.

4. What is pi??
A number can be placed into several categories based on
its properties. Is it prime or composite? Is it imaginary or
real? Is it transcendental or algebraic? These questions
help to define a number's behaviour in different situations.
In order to understand where π fits in to the world of
mathematics, one must understand several of its
properties: π is irrational and π is transcendental.
Another important concept to understand is that of how π
is calculated and how the methods have changed over
time.
π is:-
"1: the 16th letter of the Greek
alphabet...
2 : the symbol pi denoting the
ratio of the circumference of a
circle to its diameter.”

5.  Ever wondered what that symbol means on your
calculator. It is the first Greek letter of the Greek work meaning
“perimeter”. It turns out that it is a very important symbol.
 Pi = 3.1459.
 Some of you may already know that pi is used in a lot of
different mathematical formulas.
 As you study pi keep in mind that for thousands of years people
had to calculate pi by hand. This took an incrediably long time.

6. Pi 3.14159
Pi is the mathematical constant whose value is the ratio of a circle’s
circumference to its diameter.
Circumference = The distance around a circle
Diameter = The width of a circle.
~=

7. The early Babylonians and
Hebrews used “3”as a value
for Pi.
Later, Ahmed, an Egyptian
found the area of a circle .
Down through the ages,
countless people have
puzzled over this same
question, “What is Pi?
The Greeks found Pi to be
related to cones, ellipses,
cylinders and other geometric
figures.

8. LEIBNITZ (1671) Pi= 4(1/1-1/3+1/5-1/7+1/9-
1/11+1/13+...)
WALLIS Pi= 2(2/1*2/3*4/3*4/5*6/5*6/7*...)
MACHIN (1706) Pi=16(1/5- 1/(3+5^3) +1/(5+5^5) -
1/(7+5^7)+...) x
- 4(1/239 -1/(3*239^3) + 1/(5*239^5)-
...)
SHARP (1717) Pi= 2*Sq.Rt(3)(1-1/3*3 + 1/5*3^2 -
1/7*3^5...)
EULER (1736) Pi= Sq.Rt(6(1+1/1^2+1/2^2+ 1/3^2...))
BOUNCKER Pi= 4 --- 1+1 --- 2+9 --- 2+25 +...

9. THE DISCOVERY OF PI
Who? When? Discovery Equivalence
Egyptians 2000 BC (4/3)4
3.160493827...
Babylonians 2000 BC 3 1/8 3.125
Indians 2000 BC
square
root of 10 3.16227766...
Archimedes 250 BC 22/7 3.14128…
Computers Today Pi 3.1415926535…
Currently the value of pi is known to 6.4 billion places!

10. Pi slept for nearly 1500 years

11. Who Discovered Pi?
Archimedes was the
first to theoretically
approximate pi
He calculated that pi
was “trapped” between
223/71 and 22/7, or
roughly 3.1428
Today we use better
approximations, most
of which are derived by
computers
Hi, I’m
Archimedes…
Pi = Between
223/71 and 22/7

12. The Death of Archimedes
Archimedes was born in 287 BC in
a Greek state called Syracuse,
Sicily.
The city of Syracuse was taken
over by the Romans and
Archimedes was killed.
It is said that he was busy drawing
circles in the dust and writing
mathematical equations at the time
of his death

13. Pi in everyday life
 We use it for Drawing, machining,
plans, planes, buildings,
bridges, geometry problems, radio, TV,
radar, telephones, estimation, testing,
simulation, global paths, global
positioning, space science, orbit
calculation, Space ships, satellites,
Speedometers at vehicles… etc.
 Pi is used by every career whether you
are a electrical engineer, statistician,
biochemist, or physicist. Pi is indeed a
necessity for life.

14. The Usefulness of Pi
 Pi is extremely useful in calculating the area and circumference
of a sphere: A = πr2 and C = 2πr.
 Many disciplines of science use π in their equations to describe
the world
 In fact DNA, rainbows, the human eye, music, color, and
ripples all have some natural roots in pi.

15. Pi in the Professions.
 Agricultural professionals may use pi to determine the
area covered by a pivot irrigation system or storage
facility. The would use the formula
Architects and construction works would both use the formula
for Area extensively. They also use the formula for volume
extensively to fill columns of concrete and to know the space
a building would take up.

16. Pi facts
 ”Pi Day” is celebrated on March 14. The
official celebration begins at 1:59 p.m. to
make an appropriate
3.14159 when combined with the date.
 Albert Einstein was born on Pi Day
(14/3/1879) in Ulm Wurttemberg,
Germany.
 Pi goes on for ever
 Decimals have no pattern and don’t
repeat.

17. .
Pi Pie at Delft University

19. HOW TO FIND PI
The
circumference of
the circle is
smaller than the
perimeter of the
outer hexagon,
and bigger than
the perimeter of
the inner
hexagon.

20. The perimeter of the inner hexagon is
exactly 6 radii.
The perimeter of the outer hexagon is
harder to work out.
The length of each side
is 2R Tan 30O

21. The length of each side is 2R Tan
30O
So, the perimeter of the outer
hexagon is
6 x 2R Tan 30O = 12R Tan 30O
6R < Circumference < 12RTan 30O
C = 2 p R
3 < p < 3.4

22. sides p min p max
6 3.000000 3.464102
12 3.105829 3.215390
24 3.132629 3.159660
48 3.139350 3.146086
96 3.141032 3.142715
192 3.141452 3.141873
384 3.141558 3.141663
768 3.141584 3.141610
1,536 3.141590 3.141597
3,072 3.141592 3.141594
6,144 3.141593 3.141593
In general, for an n-sided polygon inside the
circle and another one outside of the circle,
n x Sin (180/n) < p < n x Tan (180/n)

23. Summary
 Pi is the mathematical constant whose value is the ratio of a circle’s
circumference to its diameter.
 It has taken 100’s of different algorithms to help estimate pi’s number
because it is a repeating decimal.
 Pi has been developed along with the development of mathematics.
 There are many different applications for pi.
 Many different professionals such as engineers, agriculturalist, and
construction workers use pi.
 Pi day is celebrated on 3/14 of every year because pi = 3.14 

25.  The mathematical constant e is the base of
the natural logarithm (ln(x) or loge(x)).
 It is often called Euler's number, due to the
related and extensive discoveries of Euler.
The exact reasons why Euler himself started
to use the letter e for the constant are
unknown, but it may be because it is the first
letter of the word "exponential" rather than of
his name, because he was a very modest
man.
Euler's number
e＝2,7182818284590452...

26. History
 Jacob Bernoulli, another famous Basel mathematician, had defined
the number e as that he estimated between 2 and 3.
 He was conscious of the unproved equivalence with the 2 other
formulas:

27. Mathematical properties
The exponential function f(x) = ex is
important in part because it is the
unique nontrivial function (up to
multiplication by a constant) which is
its own derivative, and therefore, its
own primitive:

28. Euler's formula of complex analysis
 Euler’s identity is an equality found in mathematics that has
been compared to a Shakespearean sonnet and described
as "the most beautiful equation."
 The comple x analysis equation eix = cos + i sinx is
called Euler's identity. The special case with x = π is the
famous Euler's formula:

29. e in physics and engineering
 Complex numbers are used in almost every field of physics and
engineering.
 Euler's formula is widely used in quantum mechanics, relativity.
 In electrical engineering, signals that vary periodically over time are
often described as a combination of sine and cosine functions, and are
more conveniently expressed as the real part of exponential functions
with imaginary exponents, using Euler's formula. The Fourier series
are based on it.
 In fluid dynamics, Euler's formula is used to describe potential flow.
Concrete examples of problems include the calculation of optimal
shapes of transport vehicles, of energy generators, of boat turbines,
the simulation of chemical/physical processes, the weather forecast,
etc.
 In differential equations, the function eix is often used to simplify
derivations, even if the final answer is a real function involving sine
and cosine. Euler's identity is an easy consequence of Euler's formula.

30. e in finance
e is also the amount of money if you
deposit 1 Swiss Frank at a continuously
compounded interest rate of 100%: the
terminal value of an interest
compounded m times per annum is (1
+ 100%/m)m , and it tends to
e=2.71828... if m →∞.

31. e in operations research:
the "optimal stopping problem"

32. e in probability and statistics
The normal distribution ,describing
measurement errors and that is used daily in
statistics, was developed based on certain
discoveries of euler.

33. The function f(x) = ex
and its derivative
Euler made very important
contributions to complex analysis,
with the e number. He discovered
what is now known as Euler's
formula, i.e. that for any real
number x, the complex
exponential function satisfies
$2,718,281,828
Google in 2004 announced
its intention to raise
$2,718,281,828 stock.
eix = cos + i sinx

34. FORMULAS RELATED TO E

35. The natural log at (x-
axis) e, ln(e), is equal to 1
The area between the x-axis and the
graph y = 1/x, between x =
1and x = e is 1.
WAYS TO SHOW E:

36. E SONG

37. REFERENCES
https://en.wikipedia.org
https://www.youtube.com
PLANE GEOMETRY 10
http://euler-2007.ch
https://en.wikipedia.org/wiki/Euler