A beautiful presentation describing the history of pi and its use and application in real life situations. It also covers calculating pi and world records about the number of digits of pi that have been calculated. Hope you enjoy and use it!!

2. Mathematics
A Project on Pi

3. Certificate of Accomplishment
This is to certify that Manul Goyal, a student of class X Sarabhai
has successfully completed the project on the topic A Project on Pi
under the guidance of Amit Sir (Subject Teacher) during the year
2015-16 in partial fulfillment of the CCE Formative Assessment.
Signature of Mathematics Teacher

4. Contents
Introduction
What
is
Pi?
History
Calculating
Pi
History
of
Calculating
Pi
Buffon’s
Needle
Method
Properties
of Pi
Digits
of
Pi
Irrationality
of
Pi
Facts about
Pi
Pi
and
the Area
of Circles
Pi
in
Real Life

5. Pi is the ratio of the circumference of
a circle to the diameter.

6. What is Pi (π)?
› By definition, pi is the ratio of the circumference of a circle to its
diameter. Pi is always the same number, no matter which circle
you use to compute it.
› For the sake of usefulness people often need to approximate pi.
For many purposes you can use 3.14159, but if you want a better
approximation you can use a computer to get it. Here’s pi to
many more digits: 3.14159265358979323846.
› The area of a circle is pi times the square of the length of the
radius, or “pi r squared”:
𝐴 = 𝜋𝑟2

7. History
› Pi is a very old number. The Egyptians and the Babylonians
knew about the existence of the constant ratio pi, although they
didn’t know its value nearly as well as we do today. They had
figured out that it was a little bigger than 3.
Community Approximation of pi
Babylonians
3
1
8
= 3.125
Egyptians
4 ×
8
9
2
≈ 3.160484
› These approximations were slightly less accurate and much
harder to work with.

8. Continued…
› The modern symbol for pi [π] was first used in our modern sense
in 1706 by William Jones, who wrote:
There are various other ways of finding the Lengths or Areas of particular
Curve Lines, or Planes, which may very much facilitate the Practice; as for
instance, in the Circle, the Diameter is to the Circumference as
1 to
16
5
−
4
239
−
1
3
16
53 −
4
2393 + ⋯ = 3.14159 … = 𝜋
› Pi (rather than some other Greek letter like Alpha or Omega) was
chosen as the letter to represent the number 3.141592… because
the letter π in Greek pronounced like our letter ‘p’ stands for
‘perimeter’.

9. 2 Pi in radians form is 360 degrees. Therefore Pi
radians is 180 degrees and 1/2 Pi radians is 90
degrees.

10. History of Calculating Pi
Archimedes
250 B.C.
Newtonand
Leibnitz
Late 17th
Century
Machin’s
Formula
Early 18th
Century
Ramanujan
20th Century
Electronic
Computers
Till Date

11. Archimedes (250 B.C.)
› The first mathematician to calculate pi with reasonable accuracy
was Archimedes, around 250 B.C., using the formula:
𝐴 = 𝜋𝑟2
› Using the area of a circle, he approximated pi by considering
regular polygons with many sides inscribed in and circumscribed
around a circle. Since the area of the circle is between the areas
of the inscribed and circumscribed polygons, you can use the
areas of the polygons (which can be computed just using the
Pythagorean Theorem) to get upper and lower bounds for the
area of the circle.

12. › Archimedes showed in this way that pi
is between 3
1
7
and 3
10
71
.
› The same method was used by the early
17th century with polygons with more
and more sides to compute pi to 35
decimal places.
› Van Ceulen did the biggest 674
calculations.
The area of the circle lies
between the areas of the
inscribed and the
circumscribed octagons.
CONTINUED…

13. Newton and Leibnitz (late 17th century)
› When Newton and Leibnitz developed calculus in the late 17th
century, more formulas were discovered that could be used to
compute pi.
› For example, there is a formula for the arctangent function:
arctan 𝑥 = 𝑥 −
𝑥3
3
+
𝑥5
5
−
𝑥7
7
+ ⋯
If you substitute 𝑥 = 1 and notice that arctan(1) is
𝜋
4
, you get a
formula for pi.

14. Machin’s Formula (early 18th century)
› Using the arctangent function to calculate pi is not useful because
it takes too many terms to get any accuracy, but there are some
related formulas that are very useful.
› The most famous of this is Machin’s formula:
𝜋
4
= 4 arctan
1
5
− arctan
1
239
› This formula and similar ones were used to push the accuracy of
approximations of pi to over 500 decimal places by the early 18th
century (this was all hand calculation!)

15. The 20th Century
› In the 20th century, there have been two important developments:
– the invention of electronic computers
– the discovery of much more powerful formulas for pi
› In 1910, the great Indian mathematician Ramanujan discovered
the following formula for pi:
1
𝜋
=
2 2
9801
𝑛=0
∞
4𝑛 ! (1103 + 26390𝑛)
(𝑛!)43964𝑛
› In 1985, William Gosper used this formula to calculate the first
17 million digits of pi.

16. Electronic Computers (since the 20th century)
› From the mid-20th century onwards, all calculations of pi have
been done with the help of calculators or computers.
› George Reitwiesner was the first person who used an electronic
computer to calculate pi, in 1949, on an ENIAC, a very early
computer. He computed 2037 decimal places of pi.
› In the early years of the computer, an expansion of pi to 100,000
decimal places was computed by Maryland mathematician
Daniel Shanks and his team at the United States Naval Research
Laboratory in Washington, D.C.
› The first 100,265 digits of pi were published in 1962.

17. Continued…
› In 1989, the Chudnovsky brothers correctly computed pi to over
1 billion decimal places on the supercomputer IBM 3090 using
the following variation of Ramanujan’s infinite series of pi:
1
𝜋
= 12
𝑘=0
∞
−1 𝑘
6𝑘 ! (13591409 + 545140134𝑘)
3𝑘 ! (𝑘!)3640320
3𝑘+3
2
› In 1999, Yasumasa Kanada and his team at the Univesity of
Tokyo correctly computed pi to over 200 billion decimal places
on the supercomputer HITACHI SR8000/MPP using another
variation of Ramanujan’s infinite series of pi. In October 2005
they claimed to have calculated it to 1.24 trillion places.

18. The 21st Century – Current Claimed World Record
› In August 2009, a Japanese supercomputer called the T2K Open
Supercomputer was claimed to have calculated pi to 2.6 trillion digits in
approximately 73 hours and 36 minutes.
› In December 2009, Fabrice Bellard used a home computer to compute
2.7 trillion decimal digits of pi in a total of 131 days.
› In August 2010, Shigeru Kondo used Alexander Yee’s y-cruncher to
calculate 5 trillion digits of pi. The calculation was done between 4 May
and 3 August. In October 2011, they broke their own record by
computing ten trillion (1013) and fifty digits using the same method but
with better hardware.
› In December 2013 they broke their own record again when they
computed 12.1 trillion digits of pi.

19. Deriving Pi: Buffon’s Needle Method
› Buffon's Needle is one of the oldest problems in the field of
geometrical probability. It was first stated in 1777.
› It involves dropping a needle on a lined sheet of paper and
determining the probability of the needle crossing one of the lines
on the page.
› The remarkable result is that the probability is directly related to
the value of pi.
The following pages will present an analytical solution to the
problem.

20. THE SIMPLEST CASE
Let's take the simple case first. In this
case, the length of the needle is one unit
and the distance between the lines is also
one unit. There are two variables, the
angle at which the needle falls (θ) and the
distance from the center of the needle to
the closest line (D). Theta can vary from
0 to 180 degrees and is measured against
a line parallel to the lines on the paper.
The distance from the center to the closest
line can never be more that half the
distance between the lines. The graph
alongside depicts this situation.
The needle in the picture
misses the line. The needle
will hit the line if the closest
distance to a line (D) is less
than or equal to 1/2 times the
sine of theta. That is, D ≤
(1/2)sin(θ). How often will
this occur?

21. CONTINUED…
The shaded portion is found with using
the definite integral of (1/2)sin(θ)
evaluated from zero to pi. The result is
that the shaded portion has a value of 1.
The value of the entire rectangle is
(1/2)(π) or π/2. So, the probability of a hit
is 1/(π/2) or 2/π. That's approximately
.6366197.
To calculate pi from the needle drops,
simply take the number of drops and
multiply it by two, then divide by the
number of hits, or 2(total drops)/(number
of hits) = π (approximately).
In the graph above, we plot D
along the ordinate and
(1/2)sin(θ) along the abscissa.
The values on or below the
curve represent a hit (D ≤
(1/2)sin(θ)). Thus, the
probability of a success it the
ratio shaded area to the entire
rectangle. What is this to
value?

22. The Other Cases
There are two other possibilities for the relationship between the
length of the needles and the distance between the lines. A good
discussion of these can be found in Schroeder, 1974. The situation
in which the distance between the lines is greater than the length of
the needle is an extension of the above explanation and the
probability of a hit is 2(L)/(K)π where L is the length of the needle
and K is the distance between the lines. The situation in which the
needle is longer than the distance between the lines leads to a more
complicated result.

23. RECORD
APPROXIMATIONS OF
PI
Given alongside is a graph
showing the historical
evolution of the record
precision of numerical
approximations to pi,
measured in decimal places
(depicted on a logarithmic
scale; time before 1400 is
not shown to scale).

24. The 'famous five' equation connects the five most
important numbers in mathematics, viz., 0, 1, e, π,
and i: 𝒆𝒊𝝅 + 𝟏 = 𝟎.

25. Digits of Pi
› Pi is an infinite decimal, i.e., pi has infinitely many numbers to
the right of the decimal point. If you write pi down in decimal
form, the numbers to the right of the 0 never repeat in a pattern.
Although many mathematicians have tried to find it, no repeating
pattern for pi has been discovered - in fact, in 1768 Johann
Lambert proved that there cannot be any such repeating pattern.
› As a number that cannot be written as a repeating decimal or a
finite decimal (you can never get to the end of it) pi
is irrational: it cannot be written as a fraction (the ratio of two
integers).

26. Irrationality of Pi
Here's a proof of the irrationality of Pi from Robert Simms:
Theorem: Pi is irrational.
Proof: Suppose 𝜋 =
𝑝
𝑞
, where p and q are integers. Consider the
functions 𝑓𝑛(𝑥) defined on [0, π] by
𝑓𝑛 𝑥 =
𝑞 𝑛
𝑥 𝑛
(𝜋 − 𝑥) 𝑛
𝑛!
=
𝑥 𝑛
(𝑝 − 𝑞𝑥) 𝑛
𝑛!
Clearly 𝑓𝑛 0 = 𝑓𝑛 𝜋 = 0 for all n. Let 𝑓𝑛[𝑚](𝑥) denote the mth
derivative of 𝑓𝑛(𝑥). Note that 𝑓𝑛 𝑚 0 = −𝑓𝑛 𝑚 𝜋 = 0 for
𝑚 ≤ 𝑛 or for 𝑚 > 2𝑛; otherwise some integer

27. Continued…
max 𝑓𝑛(𝑥) = 𝑓𝑛
𝜋
2
=
𝑞 𝑛 𝜋
2
2𝑛
𝑛!
By repeatedly applying integration by parts, the definite integrals
of the functions 𝑓𝑛 𝑥 sin 𝑥 can be seen to have integer values. But
𝑓𝑛 𝑥 sin 𝑥 are strictly positive, except for the two points 0 and pi,
and these functions are bounded above by
1
𝜋
for all sufficiently
large n. Thus for a large value of n, the definite integral of
𝑓𝑛 𝑥 sin 𝑥 is some value strictly between 0 and 1, a contradiction.

28. Pi day is celebrated on March 14 at the
Exploratorium in San Francisco (March 14 is 3/14).

29. PI AND THE AREA
OF CIRCLES
› The distance around a circle is called
its circumference. The distance across
a circle through its center is called
its diameter.
› The diameter of a circle is twice as long
as the radius (𝑑 = 2𝑟).
› We use the Greek letter π to represent
the ratio of the circumference of a circle
to the diameter. For simplicity, we use
𝜋 = 3.14.
› The formula for circumference of a
circle is: 𝐶 = 𝜋𝑑, i.e., 𝐶 = 2𝜋𝑟.
Parts of a circle.

30. CONTINUED…
Square units with area 1
cm2 each inside a circle.
› The area of a circle is the number of
square units inside that circle.
› If each square in the circle to the left
has an area of 1 cm2, you could count
the total number of squares to get the
area of this circle. Thus, if there were a
total of 28.26 squares, the area of this
circle would be 28.26 cm2.
› However, it is easier to use one of the
following formulas:
𝐴 = 𝜋𝑟2
or 𝐴 = 𝜋. 𝑟. 𝑟
where 𝐴 is the area, and 𝑟 is the radius.

31. Pi in Real Life
› Pi is used in areas ranging from geometry to probability to
navigation. Common real-world application problems involve
finding measurements of circles, cylinders, or spheres, such as
circumference (one-dimensional), area (two-dimensional), or
volume (three-dimensional). Students can look for real-life
applications of knowing measurements of objects in these shapes.

32. CONTINUED…
› For example, if a student would decide
to build a robot, he might want to be
able to program the robot to move a
certain distance. If the robot is made
with wheels that have a three-inch
diameter, how far would the robot move
with each complete rotation of the
wheels? This would be a circumference
problem, as the robot would move the
length of the circumference with each
rotation of its wheels – in this case, 5π,
or approximately 15.7 inches.
Applications of Pi.

33. πManul Goyal
πRoll No. 20
πClass X
17th June’15 MLZS