Tumain

Nicholas

MAT

310

December

11, 2017.

LIFE

OF PI (?)

Introduction

Pi, in mathematics, is the ratio of

the circumference of a circle to its diameter. Thus, any circle divided by its

diameter yields this unique number. There is not an exact time to trace when it

was invented as there is limited documentation on it from various

mathematicians. ? is an irrational number, which means that its value cannot be

expressed exactly as a fraction m/n, where m and n are integers. Consequently,

its decimal representation never ends or repeats. It is also a transcendental

number, which implies, among other things, that no finite sequence of algebraic

operations on integers (powers, roots, sums, etc.) can be equal to its value.

Throughout the history of mathematics, there has been much effort to determine

? more accurately and to understand its nature; fascination with the number has

even carried over into non-mathematical culture. The Greek letter ?, often

spelled out pi in text, was adopted for the number from the Greek word for

perimeter “??????????”, first by William Jones in 1707, and

popularized by Leonhard Euler in 1737. Pi is commonly recognized as an

approximation of 22/7. It is often used for everyday calculations making it one

of the most applicable approximations in various fields of study. To 39 decimal

places, pi can be written as 3.141592653589793238462643383279502884197. In this

essay, intend to discuss briefly its history and a few methods of approximations

of pi from different mathematicians.

History of Pi

In the long history of the number ?,

there have been many twists and turns, many inconsistencies that reflect the

condition of the human race as a whole. Through each major period of world

history and in each regional area, the state of intellectual thought, the state

of mathematics, and hence the state of ?, has been dictated by the same

socio-economic and geographic forces as every other aspect of civilization (Beckmann,1974).

The following is a brief history, organized by period and region, of the

development of our understanding of the number ?.

In ancient times, ? was

discovered independently by the first civilizations to begin agriculture. Their

new sedentary life style first freed up time for mathematical pondering, and

the need for permanent shelter necessitated the development of basic

engineering skills, which in many instances required a knowledge of the

relationship between the square and the circle. Although there are no surviving

records of individual mathematicians from this period, historians today know

the values used by some ancient cultures. Here is a sampling of some cultures

and the values that they used: Babylonians – 3 1/8, Egyptians – (16/9)^2,

Chinese – 3, Hebrews – 3 (implied in the Bible, I Kings 7:23).

The first record of an

individual mathematician taking on the problem of ? (often called

“squaring the circle,” and involving the search for a way to clearly

relate either the area or the circumference of a circle to that of a square)

occurred in ancient Greece in the 400’s B.C. (this attempt was made by

Anaxagoras) (Morris 1981). Based on this fact,

it is not surprising that the Greek culture was the first to truly delve into

the possibilities of abstract mathematics. The part of the Greek culture centered

in Athens made great leaps in the area of geometry, the first branch of

mathematics to be thoroughly explored. Antiphon, an Athenian philosopher, first

stated the principle of exhaustion. Hippias of Elis created a curve called the

quadratrix, which actually allowed the theoretical squaring of the circle,

though it was not practical (Beckmann,1974).

In the late Greek period (300’s-200’s B.C.),

after Alexander the Great had spread Greek culture from the western borders of

India to the Nile Valley of Egypt, Alexandria, Egypt became the intellectual

center of the world. Among the many scholars who worked at the University

there, by far the most influential to the history of ? was Euclid. Through the

publishing of Elements, he provided countless future mathematicians with the

tools to tackle the ? problem. The other great thinker of this time,

Archimedes, studied in Alexandria but lived his life on the island of Sicily.

It was Archimedes who approximated his value of ? to about 22/7, which is still

a common value today (Beckmann,1974). Archimedes was killed in 212 B.C. in

the Roman conquest of Syracuse. In the years after his death, the Roman Empire

gradually gained control of the known world. Despite their other achievements,

the Romans are not known for their mathematical achievements. The dark period

after the fall of Rome was even worse for ?. Little new was discovered about ?

until well into the decline of the Middle Ages, more than a thousand years

after Archimedes’ death. While ? activity stagnated in Europe, the situation in

other parts of the world was quite different.

The Mayan civilization, situated on the

Yucatan Peninsula in Central America, was quite advanced for its time. The

Mayans were top-notch astronomers, developing a very accurate calendar. In

order to do this, it would have been necessary for them to have a fairly good

value for ?. Though no one knows for sure (nearly all Mayan literature was

burned during the Spanish conquest of Mexico), most historians agree that the

Mayan value was indeed more accurate than that of the Europeans (Rafael,1998).

The Chinese in the 5th century calculated ? to an accuracy not surpassed by

Europe until the 1500’s. The Chinese, as well as the Hindus, arrived at ? in

roughly the same method as the Europeans until well into the Renaissance, when

Europe finally began to pull ahead.

During the Renaissance period,

? activity in Europe began to finally get moving again. Two factors fueled this

acceleration: the increasing importance of mathematics for use in navigation,

and the infiltration of Arabic numerals, including the zero and decimal

notation. Leonardo Da Vinci and Nicolas Copernicus made minimal contributions

to the ? endeavor, but François Viète made significant improvements to

Archimedes’ methods (Beckman, 1971). The efforts of Snellius, Gregory, and John

Machin eventually culminated in algebraic formulas for ? that allowed rapid

calculation, leading to ever more accurate values of ? during this

period.

In the 1700’s the invention

of calculus by Sir Isaac Newton and Leibniz rapidly accelerated the calculation

and theorization of ?. Using advanced mathematics, Leonhard Euler found a

formula for ? that is the fastest to date. In the late 1700’s Lambert (Swiss)

and Legendre (French) independently proved that ? is irrational. Although

Legendre predicted that ? is also transcendental, this was not proven until

1882 when Lindemann published a thirteen-page paper proving the validity of

Legendre’s statement. Also in the 18th century, George Louis Leclerc, Comte de

Buffon, discovered an experimental method for calculating ?. Pierre Simon

Laplace, one of the founders of probability theory, followed up on this in the

next century. Buffon’s Needle was first posed by Comte de Buffon in 1777. He

stated that if there existed a plane, ruled with parallel lines D distance from

each other, and one was to drop a needle of length D on to that plane, the

probability that the needle would cross one of the lines was exactly 2/?. (Beckman,

1971)

Methods for approximating

Pi

There are many people who have

discovered and proved what pi is. As time goes on people discover more and more

of the seemingly random numbers. Four of

the people who proved pi are the Liu Hui, Archimedes of Syracuse, James

Gregory, and the Bible.

Liu Hui was a Chinese mathematician whose

method for proving pi was to find the area of a polygon inscribed in a circle.

When the number of sides on the inscribed polygon increased, its area became

closer to the circumference of a circle and pi. For finding the side length of

an inscribed polygon Liu Hui used a simple formula. (13Ma3) To find the side

length of an inscribed polygon of 2n sides, if the side length of a polygon with

n sides is known he used the following formula:

k6 = 1

k2n =

Sn =

In

this formula k stands for a temporary

variable, and Sn stands

for the side length of an inscribed polygon with n sides. Starting with a hexagon inside of a circle. The radius of

the circle is one, the area is pi. The side length of the hexagon is 1. To

calculate the next k value, all we

need to do is do an addition and a square root like in the following:

k6 = 1

S6 = = 1

k12 = S12 = = 0.518

k24 =

S24 = =0.261

The area of a regular polygon is

A=1/2nsa. The n stands for number of

sides, s stands for side length, and a stands for apothem. As the number of

sides increases, the apothem becomes closer and closer to the radius so we let

a =1. We now have the formula for the area of a polygon with n sides. This

formula is Pn =1/2nSn. In this formula Pn represents the area of a polygon with

n sides. Going through the formula,

one will get pi and eight of its decimal places when there are 98,304 sides,

which will give you 3.14159265.

The method Archimedes used for finding

pi was to take the perimeters of polygons inscribed and circumscribed about a

given circle. However instead of trying to measure the polygons one by one, he

used a theorem Euclid created to develop a numerical procedure for finding the

perimeter of a circumscribing polygon with 2n sides, once the perimeter of the

polygon with n sides is known (Beckman, 1971).

Then beginning with a circumscribed

hexagon, he used the formula to find the perimeters of circumscribing polygons

of 12, 24, 48, and 96. He then repeated this after developing a corresponding

formula for inscribed polygons.

Here

is the procedure.

Let

AB represent the diameter of any circle, O being its center, AC the tangent at

A; and let the angle AOC be one-third of a right angle. (13Ma2)

OA:

AC>265:153 OC:

AC=306:153

First

draw OD bisecting AOC and meeting AC in D.

Co:

OA=CD: DA

(CO+OA):

CA=OA: AD

OA:

AD>571:153

OD^2:

AD^2> 349450: 23409

OD:

DA>591 1/8: 153

Secondly,

let OE bisect the angle AOD, meeting AD in E.

OA:

AE>1162 1/8: 153

OE:

EA>472 1/8: 153

Thirdly,

let OF bisect the angle AOE and AE and F. This results in:

OA:

AF> 2334 1/4: 153

OF:

FA> 2339 1/4: 153

Fourthly,

let OG bisect the angle AOF meeting AF in G. This will give us:

OA:

AG> 4673 1/2: 153

AB=2OA GH=2AB

AB

🙁 perimeter of a polygon of 96 sides) > 4673 1/2: 14688

14688

= 3+667 ½