π (pi) is the ratio of a circle's circumference to its diameter. It is an irrational and transcendental number represented by the Greek letter π. William Jones is believed to have first used π in its modern sense in 1706 to represent the constant ratio, rather than a varying circumference. Important formulas using π include the circumference of a circle (2πr), area of a circle (πr^2), and volume/surface area of a sphere. While π cannot be expressed exactly as a fraction, it is commonly approximated as 22/7 or 3.14159.

1. SUMMER PROJECT WORK
VIKASH RESIDENTIAL
SCHOOL
MATHEMATICS PROJECT
WORK

2. PRESENTED TO :-
URMILLA MISS
PRESENTED BY :-
KAUSHIKI PATRA
IX A 09

3. INVENTION AND ROLE OF PIE
 The number π is a mathematical constant, the ratio of
a circle's circumference to its diameter, commonly
approximated as 3.14159. It has been represented by the Greek
letter "π" since the mid-18th century, though it is also
sometimes spelled out as "pi“
Name
 The symbol used by mathematicians to represent the ratio of a
circle's circumference to its diameter is the lowercase Greek
letter π, sometimes spelled out as pi. In English, π is
pronounced as "pie" In mathematical use, the lowercase
letter π (or π in sans-serif font) is distinguished from its
capital counterpart Π, which denotes a product of a sequence.

4. Definition
 π is commonly defined as the ratio of
a circle's circumference C to its diameter d
The ratio C/d is constant, regardless of the circle's size. For
example, if a circle has twice the diameter of another circle
it will also have twice the circumference, preserving the
ratio C/d.

5. Properties
 π is an irrational number, meaning that it cannot be
written as the ratio of two integers (fractions such
as 22/7 are commonly used to approximate π; no common
fraction (ratio of whole numbers) can be its exact
value). Since π is irrational, it has an infinite number of
digits in its decimal representation, and it does not settle
into an infinitely repeating pattern of digits. There are
several proofs that π is irrational.
 More strongly, π is a transcendental number, which means
that it is not the solution of any non-
constant polynomial with rational coefficients.

6. Decimal representation
of pie
The digits of π have no
apparent pattern and have
passed tests for statistical
randomness, including
tests for normality; a
number of infinite length
is called normal when all
possible sequences of digits
(of any given length)
appear equally often.

7. History of Pie
 The history of the constant ratio of the circumference to
the diameter of any circle is as old as man's desire to
measure; whereas the symbol for this ratio known today as
π (pi) dates from the early 18th century. Before this the
ratio had been awkwardly referred to in medieval Latin as:
quantitas in quam cum multiflicetur diameter, proveniet
circumferencia (the quantity which, when the diameter is
multiplied by it, yields the circumference).
It is widely believed that the great Swiss-born mathematician
Leonhard Euler (1707-83) introduced the symbol π

8. Fig:-LeonhardEuler
(1707-83)
into common use. In fact it was first
used in print in its modern sense in
1706 a year before Euler's birth by a
self-taught mathematics teacher
William Jones (1675-1749) in his
second book Synopsis Palmariorum
Matheseos, or A New Introduction to
the Mathematics based on his
teaching notes.

9. Fig:-WilliamJones
Before the appearance of the
symbol π, approximations such as
22/7 and 355/113 had also been
used to express the ratio, which
may have given the impression
that it was a rational number.
Though he did not prove it, Jones
believed that π was an irrational
number: an infinite, non-
repeating sequence of digits that
could never totally be expressed
in numerical form.

10. In Synopsis he wrote: '... the exact proportion between the
diameter and the circumference can never be expressed in
numbers...'. Consequently, a symbol was required to
represent an ideal that can be approached but never
reached. For this Jones recognised that only a pure platonic
symbol would suffice .
The symbol π had been used in the previous century in a
significantly different way by the rector and
mathematician, William Oughtred (c. 1575-1 660), in his
book Clavis Mathematicae (first published in 1631).
Oughtred used π to represent the circumference of a given

11. Fig:- WilliamOughtred
( 1575-1660 )
circle, so that his π varied according
to the circle's diameter, rather than
representing the constant we know
today. The circumference of a circle
was known in those days as the
'periphery.
On Oughtred's death in 1660 some
books and papers from his fine
mathematical library were acquired by
the mathematician John Collins (1625-
83), from whom they would eventually
pass to Jones.

12. Prove that pie was a irrational number
 The irrationality of π was not proved until 1761 by
Johann Heinrich Lambert (1728-77), then in 1882
Ferdinand Lindemann (1852-1939) proved that π was a
non-algebraic irrational number, a transcendental
number (one which is not a solution of an algebraic
equation, of any degree, with rational
coefficients.

13. Use
 Because π is closely related to the circle, it is found in many
formulae from the fields of geometry and trigonometry, particularly
those concerning circles, spheres, or ellipses. Formulae from other
branches of science also include π in some of their important
formulae, including sciences such as statistics, fractals,
thermodynamics, mechanics, cosmology, number theory, and
electromagnetism.
Geometry and trigonometry
π appears in formulae for areas and volumes of geometrical shapes
based on circles, such as ellipses, spheres, cones, and torus. Below
are some of the more common formulae that involve π.

14. I . The circumference of a circle with
radius r is 2πr.
II . The area of a circle with
radius r is πr2.
III . The volume of a sphere with
radius r is 4/3πr3.
IV . The surface area of a sphere with
radius r is 4πr2.

15. Outside Mathematics
Describing Physical Phenomena
Although not a physical constant, π appears routinely in equations
describing fundamental principles of the universe, often because
of π's relationship to the circle and tospherical coordinate
systems. A simple formula from the field of classical mechanics
gives the approximate period T of a simple pendulum of
length L, swinging with a small amplitude (g is the earth's
gravitational acceleration)
MemorizingDigits
Many persons have memorized large numbers of digits of π, a
practice called piphilology. One common technique is to
memorize a story or poem in which the word lengths

16. represent the digits of π: The first word has three letters, the second
word has one, the third has four, the fourth has one, the fifth has
five, and so on. An early example of a memorization aid, originally
devised by English scientist James Jeans, is "How I want a drink,
alcoholic of course, after the heavy lectures involving quantum
mechanics." When a poem is used, it is sometimes referred to as
a piem. Poems for memorizing π have been composed in several
languages in addition to English.
The record for memorizing digits of π, certified by Guinness World
Records, is 67,890 digits, recited in China by Lu Chao in 24
hours and 4 minutes on 20 November 2005. In 2006, Akira
Haraguchi, a retired Japanese engineer, claimed to have recited
100,000 decimal places, but the claim was not verified by
Guinness World Records

17. In popular culture
Perhaps because of the simplicity of its definition and its
ubiquitous presence in formulae, π has been represented in
popular culture more than other mathematical constructs.
In the 2008 Open University and BBC documentary co-
production, The Story of Maths, aired in October 2008 on BB
Four, British mathematician Marcus du Sautoy shows
a visualization of the - historically first exact - formula for
calculating the π when visiting India and exploring its
contributions to trigonometry.
During the 2011 auction for Nortel's portfolio of valuable
technology patents, Google made a series of unusually
specific bids based on mathematical and scientific
constants, including π.

18. Many schools in the United States observe Pi Day on 14 March
(written 3/14 in the US style). π and its digital representation are
often used by self-described "math geeks" for inside jokes among
mathematically and technologically minded groups. Several
college cheers at the Massachusetts Institute of Technology include
"3.14159". Pi Day in 2015 was particularly significant because the
date and time 3/14/15 9:26:53 reflected many more digits of pi.