Pi Essay, Research Paper
The History of Pi
A little known verse in the bible reads ?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 about(I Kings 7, 23).? 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 it?s 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 ?The Birds?. 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-chi?h who approximated pi at 355/113 in 480 AD. Next to nothing except for this work is known about Tsu Chung-chi?h?s 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 Al?Khwarizimi 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 2?s 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 Archimedes?s method of calculating pi, and Snell?s 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 it?s 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 Lambert?s 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 ?the lightning calculator? found pi to 200 decimal places. Shanks soon overshadowed Dase?s 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 Shank?s 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 it?s 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 it?s 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 1980?s 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 Kanada 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. Kanada 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 today?s standards, however still impressive in its own way.