Essay, Research Paper: Pi Number

Mathematics

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A little known verse in the bible reads ďAnd 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 about (I Kings 7, 23).Ē
This passage from the bible demonstrates the ancient nature of the irrational
number pi. Pi in fact is mentioned in a number of verses throughout the bible.
In II Chronicles 4,2, in the passage describing the building of the great temple
of Solomon which was built around 950BC, pi is given as equal to three. This
value is not very accurate at all and should not even be considered accurate for
itís time, however it should be noted that precision was not needed for the
task that was being performed and we should let the general concept of pi that
the biblical characters posses impress us. Present knowledge suggests that the
concept of pi first developed in 2000 BC in two separate cultures. The
Babylonians used pi at a value of 25/8 while an entirely different culture, the
ancient Egyptians used pi at a value of 256/81. While the biblical calculation
of pi=3 most likely came from crude measurement, there is strong reason to
believe, because of the relative accuracy of the values, that the Babylonians
and Egyptians found pi by means of mathematical equations. In the Egyptian Rhind
Papyrus, which is dated around 1650 BC, there is strong evidence supporting that
the Egyptians used 4(8/9)2 =3.16 for their value of pi. At that point in
history, and for the majority of modern history, pi was not seen as an
irrational number as it is today. The next culture that investigated pi was the
ancient Greeks. Starting in 434 BC Greeks were unraveling the mysteries of pi.
The mathematician Anaxagoras made an unsuccessful attempt at finding pi, which
he called squaring the circle and in 414 BC, 20 years after Anaxagoras failed in
his attempt to square the circle, Aristophanes refers to the work of Anaxagoras
in his comedy ďThe BirdsĒ. It took over 100 years for the Greeks to finally
find a value for pi. In 240 BC Archimedes of Syracuse showed that
223/71*pi*22/7. Archimedes knew, what so many people today do not, that pi does
not equal 22/7 and he made no claim to have discovered the exact value of pi.
However if we take the average of his two bounds we obtain pi=3.1418, which was
an error of about 0.0002. Archimedes found the most accurate value of pi up to
that time and his value would be used exclusively until the next discovery in
the world of pi. The next major finding concerning pi did not occur in the
western world, but in China by Tsu Chung-chiíh who approximated pi at 355/113
in 480 AD. Next to nothing except for this work is known about Tsu Chung-chiíhís
life but it is very unlikely that he had any awareness of Archimedes work. We
shall now notice how during the dark ages of Europe, the lead in the research of
pi is passed to the East. Aryabhata, working on his own in Persia without any
outside information in 515 AD was able to approximate pi to 3 decimal places. A
mathematician from Baghdad named AlíKhwarizimi worked with pi however the most
accurate finding of pi to date was found even more east in Samarkand by Al-Khashi.
In 1430 AD he approximated pi to 16 decimal places, the most to date. His work
however, would be the last of note from the east as the European Renaissance
brought about a whole new mathematical world. The first notable discovery in the
approximation of pi from the European Renaissance was by Viete in 1593 AD. He
expressed pi as an infinite product by using only 2ís and square roots. In
1610 Ludolph van Ceulen demonstrated the new thought coming out of the
Renaissance by calculating pi to 35 decimal places. Around the same time, Snell
refined Archimedesís method of calculating pi, and Snellís work was used by
Grienberger to calculate pi to 39 decimal places in 1630. In 1655 Wallis showed
that pi/2=2/1*2/3*4/3*4/5*6/5*6/7*8/7*8/9..... The 18th centuary brought about
great achievements in the calculating of pi. In 1706, Machin found pi to 100
decimal places, the first time that feat was ever achieved and in the same year,
a British mathematician, William Jones first used pi for the circle ratio. In
1737, Euler first used the Greek letter pi to represent the mysterious number
therefore giving it itís present day name. Up until the 18th centuary, pi was
seen as a rational number, however in 1761, Lambert showed that pi was
irrational, therefore opening up a whole new world for the research of pi. Pi
became seen as a boundless number, open for limitless exploration. Soon after
Lambertís discovery, Legendre showed that pi2 is irrational. The 19th centuary
presented two mathematicians, who, without computers were able to find pi to
huge amounts of decimal places. In 1844, Johann Dase, who was described by his
contemporaries as ďthe lightning calculatorĒ found pi to 200 decimal places.
Shanks soon overshadowed Daseís findings however by finding by to an
astounding 707 decimal places in 1873. While the 19th centuary showed great
strides in the calculation of pi, the 20th centuary, with the advent of
computers, broke great barriers in finding the most exact value of pi. In 1945,
two scientists, Ferguson and Wrench worked on a computer system for calculating
pi, however before this system was perfected, they did some manual calculations.
In 1945 Ferguson found that the number occupying the 528th place for Shankís
value of pi was incorrect. Soon after in 1948, Ferguson and Wrench published the
correct value of pi to 808 decimal places. However in 1949, with their computer
up and running, Ferguson and Wrech were able to find pi to itís most exact
value ever. Their ENIAC system performed the first electronic computation of pi
to 2,037 decimal places. It is interested to not that this computer occupied a
warehouse the size of a high school fieldhouse and itís only purpose was to
calculate pi, however the computer represented a huge jump in the research of
pi. It opened doors to the intricate calculations of pi we see in our modern
day. From this point on, all new calculations of pi would be done
electronically. In 1958, Genuys found pi to 10,000 decimal places, and in 1962
David Shanks, a relative of the 19th centuary mathematician William Shanks,
along with Wrench found pi to 100,000 decimal places. In 1973 Guillard and
Bouyer were the first to find pi to one million places. The research in pi in
the 1980ís to the present has pretty much moved across the pacific to Japan.
In 1982, Y Tamura and Y Kanada found pi to 8 million places and in 1986 Kanada
found it to 33,554,000 places;in 1987 134,217,728 places and in 1988 he found pi
to 201,326,000 places. In November of 1989 Canada brought the one billion mark
by finding pi to 1,073,741,799 places. The great year of 1995 however made the
most progress in the calculation of pi. Canada found pi to 4 billion places, and
soon after Borwien, a German mathematician found pi to 10 billion places, a
great leap from the biblical approximation of 3. Today you can download files
off the internet of values of pi to 2.5 million places. On the next page you can
examine pi to 50,000 places, a relatively low number for todayís standards,
however still impressive in its own way.
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