OF PI (?)
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
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,
Methods for approximating
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
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
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.
is the procedure.
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)
draw OD bisecting AOC and meeting AC in D.
AD^2> 349450: 23409
DA>591 1/8: 153
let OE bisect the angle AOD, meeting AD in E.
AE>1162 1/8: 153
EA>472 1/8: 153
let OF bisect the angle AOE and AE and F. This results in:
AF> 2334 1/4: 153
FA> 2339 1/4: 153
let OG bisect the angle AOF meeting AF in G. This will give us:
AG> 4673 1/2: 153
🙁 perimeter of a polygon of 96 sides) > 4673 1/2: 14688
= 3+667 ½ < 3 1/7+ 3.14 The third proof I will be talking about is James Gregory's proof. James Gregory was a Scottish man, and a great mathematician. His method for proving pi was to use the Gregory series. (Sometimes called the Leibniz Series.) To find pi he had to know the area of the curve, y=1/ (1+x^2), between the 0 and the x which was the arctan x. arctan(x) = = Giving: = x – x3/3 + x5/5 – x7/7 + … So a particular example is: = tan-1 = 1 – 1/3 + 1/5 - 1/7 + … The series that Gregory was using to find pi would be accurate to four decimal places when the last omitted term is less than 0.00005. ? , or k ? 10000 If you then accelerate the series, because doing this by hand would take a long amount of time, to the thirtieth term you get pi. I will give pi and the first fourteen decimal place (Berggren, 1997). The final proof I will be talking about is the proof from the Bible. There are a few places where pi is mentioned in the Bible. The first is 1 Kings 7:23. This verse says, "And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about." The Bible also mentions pi in 2 Chronicles 4:2, it says, "Also he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about." These verses are saying that pi is equal to 3 (Blatner,1997). A cubit is the length of a man's forearm, from the elbow to the extended fingertips. It is highly probable, that if something measured less than a cubit, it was rounded and counted as a full cubit, as there is no mention of a third or fourth cubit. A Hebrew cubit was about 45 centimeters (18 inches). In the Bible it seems that the circumference was measured with a line, a string or cord on which the distance was marked, the distance then would have been measured off in cubits. The diameter would have been measured the same way as the circumference. This is what Bible scholars believed that it probably looked something like the picture below. As mentioned earlier when they measured they would round off the numbers. For example, the actual diameter was likely to be 9.65 which would have been rounded to 10 cubits. The actual circumference was likely 30.32 and would have been rounded to 30. If the actual measurements were used to calculate pi, it would result to 30.32/9.65= 3.14, which is the actual measurement for pi. However since they rounded the cubits they would have gotten a different answer. Their diameter would have been 10. The circumference would have been 30. This would have given them the equation 30/10= 3. This is what was used as pi. While this is not completely accurate it is quite close. Conclusion Pi is closely, and always has been, related to the circle. It is used in trigonometry, geometry and with anything using circles or spheres. Since Pi is being infinitely calculated, it is believed, in theory, that Pi contains every bit of information to ever exist. The numbers being calculated by Pi are always expanding, therefore creating a larger and larger chance for something to be written in those numbers. Works Cited Beckmann, Petr. A History of Pi (Pi). New York : St. Martin's, 1974 c1971., 1974. EBSCOhost, research.westminster-mo.edu:2048/login?url=http://search.ebscohost.com/login.aspx?direct=true=cat02498a=westm.b1408989=eds-live. Morris, Kline. "A History of Mathematics, 3Rd Ed. Florian Cajori." American Scientist, no. 1, 1981, p. 106. EBSCOhost, research.westminster-mo.edu:2048/login?url=http://search.ebscohost.com/login.aspx?direct=true=edsjsr=edsjsr.27850324=eds-live. Schepler, Herman C. "The Chronology of PI." Mathematics Magazine, vol. 23, no. 3, 1950, pp. 165–170. JSTOR, JSTOR, www.jstor.org/stable/3029284. Berggren, Lennart, and Jonathon and Peter Borwein. Pi: A Source Book. Springer-Verlag. New York, 1997. Blatner, David. The Joy of Pi. Walker Publishing Company, Inc. New York, 1997.