Tumain Nicholas MAT 310 December 11, 2017. LIFE OF

Tumain
Nicholas

MAT
310

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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  

x

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