pi

pi

The symbol

π

was first introduced by William Jones

[

7

,

8

]

in 1706 to denote the ratio between the

perimeter

and the

diameter

on any given circle. In other words, dividing the perimeter of any circle by its diameter always gives the same answer, and this number is defined to be

π

. A 12-digit approximation of

π

is given by

3.14159265358

⁢

…

Over human history there were many attempts to calculate this number precisely. One of the oldest approximations appears in the

Rhind Papyrus

(circa 1650 B.C.) where a geometrical construction is given where

(

16

/

9

)

2

=

3.1604

⁢

…

is used as an approximation to

π

although this was not explicitly mentioned.

It wasn’t until the Greeks that there were systematical attempts to calculate

π

. Archimedes

[

1

]

, in the third century B.C. used

regular polygons

inscribed

and

circumscribed

to a circle to approximate

π

: the more sides a

polygon

has, the closer to the circle it becomes and therefore the ratio between the polygon’s area between the square of the radius yields approximations to

π

. Using this method he showed that

223

/

71

<

π

<

22

/

7

(

3.140845

⁢

…

<

π

<

3.142857

⁢

…

)

.

Around the world there were also attempts to calculate

π

.
Brahmagupta

[

1

]

gave the value of

10

=

3.16227

⁢

…

using a method

similar

to Archimedes’. Chinese mathematician Tsu Chung-Chih (ca. 500 A.D.) gave the approximation

355

/

113

=

3.141592920

⁢

…

.

Later, during the renaissance, Leonardo de Pisa (

Fibonacci

)

[

1

]

used 96-sideed regular polygons to find the approximation

864

/

275

=

3.141818

⁢

…

For centuries, variations on Archimedes’ method were the only tool known, but Viète

[

1

]

gave in 1593 the formula

2

π

=

1

2

⁢

1

2

+

1

2

⁢

1

2

⁢

1

2

+

1

2

⁢

1

2

+

1

2

⁢

1

2

⁢

⋯

which was the first analytical expression for

π

involving

infinite

summations

or products. Later with the advent of calculus many of these formulas were discovered. Some examples are
Wallis’

[

1

]

formula:

π

2

=

2

1

⋅

2

3

⋅

4

3

⋅

4

5

⋅

6

5

⁢

⋯

and Leibniz’s formula,

π

4

=

1

-

1

3

+

1

5

-

1

7

+

1

9

-

1

11

+

⋯

,

obtained by developing

arctan

⁡

(

π

/

4

)

using power series, and with some more advanced techniques,

π

=

6

⁢

ζ

⁢

(

2

)

,

found by determining the value of the

Riemann Zeta function

at

s

=

2

(

http://

planetmath

.org/ValueOfTheRiemannZetaFunctionAtS2

).

The Leibniz expression provides an alternate way to define

π

(namely 4 times the limit of the series) and it is one of the formal ways to define

π

when studying analysis in order to avoid the geometrical definition.

It is known that

π

is not a

rational number

(

quotient

of two

integers

). Moreover,

π

is not algebraic over the rationals (that is, it is a

transcendental number

). This means that no

polynomial

with rational coefficients can have

π

as a root. Its irrationality implies that its

decimal expansion

(or any integer base for that matter) is not finite nor periodic.

References

1

The MacTutor History of Mathematics archive,

http://www-groups.dcs.st-and.ac.uk/ history/BiogIndex.html

The MacTutor History of Mathematics Archive

2

The ubiquitous

π

[Dario Castellanos] Mathematics Magazine Vol 61, No. 2. April 1988.

Mathematical Association of America

.

3

A history of pi. [O’Connor and Robertson]

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_through_the_ages.html

4

Pi chronology. [O’Connor and Robertson]

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_chronology.html

5

Asian contributions to Mathematics. [Ramesh Gangolli]

http://www.pps.k12.or.us/depts-c/mc-me/be-as-ma.pdf

6

Archimedes’ approximation of pi. [Chuck Lindsey]

http://itech.fgcu.edu/faculty/clindsey/mhf4404/archimedes/archimedes.html

7

N. Higham, Handbook of writing for the mathematical sciences, Society for Industrial and Applied Mathematics, 1998.

8

The MacTutor History of Mathematics archive,

http://www-gap.dcs.st-and.ac.uk/ history/Mathematicians/Jones.html

William Jones

Title

pi

Canonical name

Pi

Date of creation

2013-03-22 11:51:41

Last modified on

2013-03-22 11:51:41

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

27

Author

mathcam (2727)

Entry type

Definition

Classification

msc 01A40

Classification

msc 01A32

Classification

msc 01A25

Classification

msc 01A20

Classification

msc 01A16

Classification

msc 51-00

Classification

msc 11-00

Classification

msc 22A22

Classification

msc 46L05

Classification

msc 82-00

Classification

msc 83-00

Classification

msc 81-00

Related topic

RegularPolygon

Related topic

Limit

Related topic

Diameter

Related topic

TranscedentalNumber

Related topic

AlgebraicNumber

Related topic

ExtensionMathbbRmathbbQIsNotFinite

Related topic

WallisFormulae

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