History of π. Andrew Dolbee. November 7, 2012

Transcription

1 History of Andrew Dolbee November 7, 01 The search for the ratio between a circle's circumference and its diameter, known as PI(), has been a very long one; appearing in some of the oldest mathematical documents ever discovered and stretching to the information age and surely beyond. While we do not know when man rst tried to calculate we do know the search was ongoing in ancient Egypt. During this time shows up in records from many places in the world; from Egypt and Babylonia to India, even showing up in the Hebrew Bible. It was not until the search reached Greece, however, that the world got its rst, recorded, algorithm for calculating. Devised by Archimedes this algorithm used polygons to approximate [1]. A similar method was used in China shortly after [1]. This polygonal approach was used up until the development of innite series, which revolutionized the search for, and are what we use, to this day, to approximate more and more digits of [1]. The next big leap in approximations came with the development of computers which allowed mathematicians to perform many more computations more accurately and more quickly than was ever possible before. The reasons for this search have changed drastically as time has gone on originally a search of necessity, in more recent years, it has become a search of leisure. While there is no concrete proof that ancient Egyptians had knowledge of during the construction of the Great Pyramid at Giza, which dates from between 1800 b.c. to 4750 b.c., the ratio of its side to its height of 11:7, making the ratio of half the perimeter to its height ( ) []. The next time shows up is in the Rhind Papyrus, one of, if not the, oldest mathematical Figure 1: Great Pyramids at Giza [9] 1

2 Figure : The Polygonal Method [10] document ever found, the creation of which dates back to 000 b.c., in which is described as 16 9 ( ) []. A Babylonian clay tablet from around the same time as the Rhind Papyrus treats as 5 8 (3.150) [1]. The last two ancient approximations of to be covered is the one found in the Shulba Sutras, from India dating to about 600 b.c. which approximated as (3.008) and a verse in the Hebrew Bible: And he made a molten sea, ten cubits from the one brim to the other : it was round al labout, and his height was five cubits: and a line of thirty cubits did compass it round about. (1 Kings 7 : 3) The passage, which describes a circular structure with a diameter of 10 cubits and a radius of 30 cubits, implies is 3. The rst recorded algorithm used to calculate was a polygonal approximation, which calculated the perimeters of polygons circumscribed and inscribed on a circle to estimate, see Figure. This method was developed by Archimedes around 50 b.c. Using this method, with 9 sided poylgons, Archimedes was able to approximate the value of to be between (3.1408) and (3.148) []. Around 65 a.d. a similar method was used in China by Liu Hui who calculated the perimeters of regular inscribed polygon of up to 19 sides and approximated as (3.1400) []. The polygonal approximation method was used for well over a thousand years and generated some very impressive results. In fact an Austrian Astronomer, Christoph Grienberger, was able to approximate as < n < using the polygonal method. [6] The rst 38 digits of this approximation are exactly correct and it remains the most accurate approximation reached with the polygonal method [1]. For over a thousand years the polygonal method dominated the search for ; it was not until the 16th and 17th century with the development of innite series that this changed. Innite series

3 Figure 3: François Viéte [11] allowed to be approximated with much greater precision than previous methods allowed [1]. The rst written algorithm using innite series is attributed to Indian astronomer Nilakantha Somayaji and were discovered around 1500 a.d. [4]. Somayaji credits Madhava of Sangamagrama of the series and they are formally referred to as the Madhava series.[9] 4 = (1) 9 French mathematician François Viéte (Figure 3) discovered Europe's rst innite series for the approximation of. François Viéte's series was an innite product, unlike most other series covered which are innite sums, which he found in 1593 [1]. + = In the mid 1660's Isaac Newton used [5] arcsin x = x + 1 x = 6( x () x , ) (3) to approximate to 15 digits [1]. In short order innite series showed their superiority to geometrical methods with Abraham Sharp, an English mathematician, using the Gregory-Leibniz series, which was, in fact, the rediscovered Madhava Series,(1) to calculate to 71 digits [1]. Then, only 7 years later, John Machin developed his own innite series, based o of the Gregory-Leibniz series, and used it to calculate to 100 digits [1]. 4 = 4 arctan 1 5 arctan 1 39 Machin's series led to the creation of many dierent series known collectively as "Machin-like formula", such as Euler's formula, [6] 4 = arctan 1 + arctan (4) (5)

4 Figure 4: The ENIAC Computer [1] that were used to set many records for the approximation of [1]. One such record was the 60-digit approximation by Daniel Ferguson in 1946 which, to this day, remains the best manual approximation of [1]. With the invention of the computer the search for was forever changed, it was now possible to run millions, and eventually billions and even trillions, of calculations a second allowing ever more digits in the approximations. In 1949 John Wrench and Levi Smith were able to approximate to 1,10 digits using a desk calculator, nearly doubling the previous record [1]. Next up was a team headed by George Reitwiesner and John von Neumann who, using the ENIAC computer (Figure 4), were able to approximate to,037 digits [7]. This and subsequent records were broken repeatedly, until in 1973, was successfully approximated to 1 million digits [1]. The reasons for calculating have been quite varied over the years. One can assume for the largest part of history these calculations were done out of necessity, such as for construction. As time went on though it is pretty easy to see that that was no longer the case, in fact as JÃ rg Arndt and Christoph Haenel state in their book, Unleashed, "39 digits of are sucient to calculate the volume of the universe to the nearest atom." [1] It is reasonable, then, to assume that at most a couple hundred digits of are all you would ever need to accurately solve any equation. This has lead to a time when the search for is done for reasons other than actually knowing, such as testing supercomputers [1]. It is also possible that some search purely for the sake of searching or a need to break records. The search for has been a long one, and, no doubt, nowhere near its end. While 10 trillion digits, the current record for number of consecutive digits of approximated [8], may seem a great feat one can just imagine our decendants, many generations from now, hearing how someone has reached the centillionith digit of. No matter where humanity treads, from the ancient pyramids to farthest reaches of space, as long as man has math we will always have 4

5 References [1] Arndt, JÃ rg; Haenel, Christoph, Pi Unleashed, Springer-Verlag (001). [] Shepler, Herman C. "The Chronology of PI", Mathematics Magazine Vol. 3, Mathematical Association of America (1950) [3] Grienbergerus, Christophorus, Elementa Trigonometrica (1630) [4] Roy, Ranjan, "The Discovery of the Series Formula for pi by Leibniz, Gregory, and Nilakantha", Mathematics Magazine Mathematical Association of America (1990) [5] Huberty, Michael D., Ko Hayashi, and Chia Vang, "Innite Expressions for Pi." (06 Jul 1997) retrieved 7 Oct 01 huberty/math5337/groupe/expresspi.html [6] Weisstein, Eric W. "Euler's Machin-Like Formula." From MathWorldA Wolfram Web Resource, retrieved 7 Oct 01 [7] Plummer, Libby. "The rst computer to calculate Pi." Pocket-Lint (14 Mar 011) retrieved 7 Oct 01 [8] Yee, Alexander J, and Shigeru Kondo, "Round Trillion Digits of Pi.", NumberWorld ( Oct 011) retrieved 17 Oct 01 [9] [10] [11] [1] 5