History of pi

- The number pi (symbol p) is a mathematical

constant that is the ratio of a circle's circumfer

ence to its diameter, and is approximately equal

to 3.14159. - It has been represented by the Greek letter "p"

since the mid-18th century, though it is also

sometimes written as pi. - p is an irrational number, which means that it

cannot be expressed exactly as a ratio of

two integers (such as 22/7) consequently,

its decimal representation never ends and

never settle into a permanent repeating pattern.

EXPLAINATION OF PI

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DEFINITION

- p 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. This definition of p implicitly makes

use of flat (Euclidean) geometry although the

notion of a circle can be extended to any

- curved (non-Euclidean) geometry, these new

circles will no longer satisfy the

formula p C/d. There are also other definitions

of p which do not mention circles at all, for

example p is twice the smallest positive x for

which cos(x) equals 0.

The circumference of a circle is slightly more

than three times as long as its diameter. The

exact ratio is called p.

Role Of Leonhard Euler In Pi

- After Jones introduced the Greek letter in 1706,

it was not adopted by other mathematicians

until Euler started using it, beginning with his

1736 work Mechanica. Before then, mathematicians

sometimes used letters as c or p instead.

Because Euler corresponded heavily with other

mathematicians in Europe, the use of the Greek

letter spread rapidly.

In 1748, Euler used p in his widely read

work Introductio in analysin infinitorum (he

wrote "for the sake of brevity we will write

this number as p thus p is equal to half the

circumference of a circle of radius 1") and the

practice was universally adopted thereafter in

the Western world.

Leonhard Euler popularized the use of the Greek

letter p in works he published in 1736 and 1748.

USES OF PI

- Because p 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 p in some of their important formulae,

including sciences such as statistics, fractals,

thermodynamics, mechanics, cosmology, number

theory, and electromagnetism.

USES OF PI

- Geometry and Trigonometry

- Probability and Statistics

- Engineering and Geology

Geometry and Trigonometry

- p appears in formulae for areas and volumes of

geometrical shapes based on circles, such

as ellipses, spheres, cones, and torus. Some of

the more common formulae that involve p - The circumference of a circle with radius r is

2pr. - The area of a circle with radius r is
- The volume of a sphere with radius r is 4/3pr3
- The surface area of a sphere with

radius r is 4pr2

The area of the circle equals p times the shaded

area.

- The trigonometric functions rely on angles, and

mathematicians generally use radians as units of

measurement. p plays an important role in angles

measured in radians, which are defined so that a

complete circle spans an angle of 2p radians. The

angle measure of 180 is equal to p radians, and

1 p/180 radians. - Common trigonometric functions have periods that

are multiples of p for example, sine and cosine

have period 2p.

Sine and cosine functions repeat with period 2p.

Probability and Statistics

- The fields of probability and statistics frequentl

y use the normal distribution as a simple model

for complex phenomena for example, scientists

generally assume that the observational error in

most experiments follows a normal

distribution. p is found in the Gaussian

function (which is the probability density

function of the normal distribution)

with mean µ and standard deviation s.

A graph of the Gaussian function ƒ(x) e-x2. The

colored region between the function and

the x-axis has area .

Engineering and Geology

- p is present in some structural engineering

formulae, such as the buckling formula derived by

Euler, which gives the maximum axial load F that

a long, slender column of length L, modulus of

elasticity E, and area moment of inertia I can

carry without buckling.

The constant p is represented in this

mosaic outside the mathematics building at

the Technische Universität Berlin.

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In Popular Culture

- Perhaps because of the simplicity of its

definition and its ubiquitous presence in

formulae, p has been represented in popular

culture more than other mathematical constructs.

Importance Given To Pi

Palais de la Découverte

Pi Day

Palais de la Découverte

- In the Palais de la Découverte (a science museum

in Paris) there is a circular room known as the

"pi room". On its wall are inscribed 707 digits

of p. The digits are large wooden characters

attached to the dome-like ceiling. The digits

were based on an 1853 calculation by English

mathematician William Shanks, which included an

error beginning at the 528th digit. The error was

detected in 1946 and corrected in 1949.

Pi Room

Pi Day

- Pi Day is an annual celebration commemorating

the mathematical constant p (pi). Pi Day is

observed on March 14, since 3, 1, and 4 are the

three most significant digits of p in the decimal

form. In 2009, the United States House of

Representatives supported the designation of Pi

Day. The earliest known official or large-scale

celebration of Pi Day was organized by Larry Shaw

in 1988 at the San Francisco Exploratorium.

Larry Shaw, the organizer of the first Pi Day

celebration at the Exploratorium in San Francisco.

Google Doodle on Pie Day

Pi Pie at Delft University

and

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SUBMITTED TO - JITENDER SIR