Finding the Value of Pi - Research Paper Example

Finding the Value of Pi Table of Contents Finding the Value of Pi Table of Contents 1. Introduction 2.History of Pi 3.Ancient Method: Archimedes’s Method for Finding the Value of π 4 4.Modern Method: Borweins’ Algorithm For Computation of π 7 5.Conclusion 8 Bibliography 9 1. Introduction The challenge posed by the ubiquitous mathematical entity called “Pi” (π) has intrigued mankind from time immemorial. Defined as the constant ratio between the circumference of a circle and its diameter, this number has drawn the attention of mathematicians of all periods in the history, mainly because it is an irrational number with a literally unending series of places after the decimal, and has graduated to the status of being a transcendental number (Sondow), thanks to all the attention it has received. The ancient mathematicians had concentrated on determining more and more precise values of Pi to as many decimals as possible. In modern times, however, with the advent of computers, the emphasis has shifted to the speed at which the value of Pi can be determined together with increasing the number of decimal places. This paper traces the history of Pi and the efforts made by mathematicians and astronomers to get closer and closer to the “precise” value of π, and then discusses two methods for determining the value of Pi – one ancient method and one modern method. 2. History of Pi 2.1. Ancient Era The very first attempts to determine the value of π date back to around 2000 B.C., when the Babylonians and Egyptians approached the problem in their own ways. While the Babylonians obtained the value of 3+1/8, the Egyptians obtained the value as (4/3) ^4 for π. About the same time, Indians used the value of square root of 10 for Pi. All these values were based, essentially, on measurement of circumferences and diameters of circles of different sizes (Beckmann, 12-15 and 98-106). The first major step towards determining the value of Pi is attributed to the great Greek mathematician and physicist, Archimedes around 250 B.C. The ancient Greeks, with their penchant for precision, were interested in precise mathematical proportions in their architecture, music and other art forms, and hence were curious about better precision in determining the value of Pi. Thus Archimedes developed a method using inscribed and circumscribed polygons for calculating better and better approximations to the value of π and came to the conclusion: 223/71 < π < 22/7 Archimedes was the first to come up with a theoretical method for determining the value of Pi, while earlier attempts were based on measurements. Subsequently, around 150 A.D., the Egyptian mathematician Ptolemy (of Alexandria) gave the value of 377/120, and around 500 A.D., the Chinese Tsu-Ch’ung-Chi gave Pi the value of 355/113. Many others like Ptolemy and Tsu-Ch’ung-Chi continued to use Archimedes’s method to calculate the vale of Pi to better approximations. Ludolph von Ceulen used this method with a 2^62-sided polygon to calculate Pi to 35 decimal places in 1596, and for this reason, Pi was referred to as the “Ludolphine number” for many years. This ubiquitous number came to be referred as “pi” only in 1706 when the English mathematician, William Jones used the Greek symbol to refer to this circle ratio, and the credit for popularizing this goes to the Swiss mathematician Leonhard Euler. 2.2. Era of Higher Math and Calculus Thus in the 17th century, with the advent of higher mathematics and the concept of calculus, several different formulas for Pi were derived. Some of the famous formulas for Pi derived by different mathematicians include: James Gregory (often credited to Gottfried Leibniz): π/4 = 1 - 1/3 + 1/5 – 1/7 +…… John Wallis: π/2 = 2*2*4*4*6*6…../1*1*3*3*5*5…. Isaac Newton: π/6 = 1/2+1/2*(1/3*2^3) + (1*3)/ (2*4)*(1/5*2^5) +... Leonard Euler: (π^2) /2 = 1/1^2 + 1/2^2 + 1/3^2 +... The fact that it was futile to search for an exact value for Pi was not known until as late as in 1761 when Johann Lambert proved that the magical Pi, was indeed irrational!!!. Subsequently in 1882, Carl von Lindemann proved that Pi was not just irrational, but was also transcendental – meaning that it is not the solution of any polynomial equation with integer coefficients. As a consequence of this proof, it was realized that: a) It is not possible to draw a square exactly equal in area to a given circle; and b) It is not possible to express π as an exact expression in surds like root5, root2/root7 etc… Thus this proof by Lindemann marked a turning point in the approach towards calculating the value of Pi. After this realization, the interest in the value of π has centered round extending the number of decimal places and in finding expressions that and approximations. The first attractive formula for finding the value of π using trigonometric functions is attributed to Machin: (Bucknall.Calculating Pi) Machin’s formula: π/2 = 4* arctan (1/5) – arctan (1/239) Using this formula, Machin managed to compute π to 100 decimal places The Indian mathematician Ramanujam, for instance, found the following approximations: (1 + (root3)/5)*7/3 = 3.14162371… (81 + (19^2)/22)^ (1/4) = 3.141592653… 63*(17 + 15*root5)/25*(7 + 15*root5) = 3.141592654… After these, the more recent algorithm that merits mention is Borwein’s quartically convergent algorithm, which is somewhat related to Ramanujam’s work on elliptic integrals (Taylor. 3.14>>History and Philosophy of Pi). 2.3. Era of Computers As time passed by, and as we reached the later half of the twentieth century, the advent of computers led to faster and more accurate calculations of π to more decimal places. Thus, in 1949 ENIAC was used to calculate π to 2037 decimal places in 70 hours, while in 1955 NORC (National Ordinance Research Center) had calculated π to 3089 decimal places in just 13 minutes!!!. The progress since then has been mind-boggling, with Y. Kanada and D. Takahashi calculating π to over 51 billion places in just over 29 hours. The latest record, as of August 2009 lies with the Japanese Supercomputer T2K-Tsukuba system calculating π to 2,576,980,370,000 (2.5+ trillion) places in 73 hours and 36 minutes (including verification time), surpassing the previous known record in 2002 by 8 times (1.2 trillion places in 600 hours). In the next Section we will look at one example of the ancient method and one recent method to find the value of π. 3. Ancient Method: Archimedes’s Method for Finding the Value of π As noted earlier, Archimedes, the Greek mathematician and physicist, was the first to use a theoretical method to determine the value of π around 250 B.C., as opposed to the earlier methods that relied largely on actual measurement of the circumference or the area of a circle, using approximations in the process. Thus his approach differed from the earlier ones in a fundamental way. Another significant improvement in the method used by Archimedes was that it was the first iterative process, whereby one can get a more accurate approximation by repeating the process, using the previous estimate of π to get a new estimate. This was a new feature in Greek mathematics, though it had an ancient tradition among the Chinese in their methods for approximating the values of square roots. Archimedes’s method is not fully documented in any available piece of literature, and hence different authors of books on the history of math have presented slight variations on his method to make it easier to understand. This paper relies on Sir Thomas Heath’s translation of the original work (14-23). The method used by Archimedes involves approximating π by the perimeters of regular polygons inscribed in and circumscribed about a circle. Consider a circle of radius 1 unit, in which we inscribe a regular polygon of 3x2n-1 sides and circumscribe a regular polygon of 3x2n-1 sides. The figure below illustrates the inscribing and circumscribing hexagons for a circle of radius = 1 unit. (Note: Here n = 2). The value of the angle AOB (the bisecting angle for the inscribed polygon) is given by π/K, where K = 3x2n-1 , which is the same as angle AOT (the bisecting angle for the circumscribing polygon) Given this construct, Let an be the semi-perimeter of the outer polygon, and Let bn be the semi-perimeter of the inner polygon. The method used by Archimedes used two basic principles: 1) Proposition 3 of Book VI of Euclid’s Elements: The full theorem states as follows: “If an angle of a triangle be bisected and the straight line cutting the angle cut the base also, the segments of the base will have the same ratio as the remaining sides of the triangle; and,(conversely) if the segments of the base have the same ratio as the remaining sides of the triangle, the straight line joined from the vertex to the point of section will bisect the angle of the triangle.” This helped in determining the values of an and bn as functions of n 2) Using the above Euclid principle, Archimedes also developed a numerical procedure for calculating the perimeter of a circumscribing polygon of 2K sides, once the perimeter of the polygon of K sides is known. Using the above two magnificent principles, Archimedes derived the following equalities: (1/an + 1/bn) = 2/an+1 (an+1*bn) = (bn+1)2 These equalities can easily be derived using trigonometric identities. But it is important to note that Archimedes did not even have the advantages of trigonometrical notations, nor did he have access to algebraic notations during his days. Despite these limitations, Archimedes calculated the values for an and bn starting with a hexagon, and using his formulas he calculated the perimeters of polygons of 12, 24, 48 and finally 96 sides. Since it is clear that the value of the circumference of the circle lies between the values of the perimeters of the inscribed and circumscribing polygons, Archimedes came to the conclusion that b6 < π < a6. In numeric terms, this translated to 223/71 < π < 22/7. 4. Modern Method: Borweins’ Algorithm For Computation of π Since the advent of the computer era, the focus in relation to efforts in determining the value of π has shifted to faster and more accurate methods leading to more digits. This has resulted in better algorithms being postulated by mathematicians around the world. While Machin was amongst the first to set the ball rolling with fundamentally different algorithms, with his formula, π/2 = 4* arctan (1/5) – arctan (1/239), and had used this to calculate π to 100 decimal places, there have been rapid advancements in algorithms. Borwein and Borwein derived a class of algorithms based on the theory of elliptical integrals that yielded very rapidly convergent approximations to elementary constants such as π, e, and root2 (Borwein et al. 562). Using this class of algorithms, they discovered a general technique for obtaining higher-order convergent algorithm for π. Thus, their quartically convergent algorithm for 1/ π can be stated as follows: If we now iterate then αn converges quartically to 1/ π. The beauty of this method is that each successive iteration approximately quadruples the number of digits in the result obtained for π. However, since this kind of computation requires a set of high-performance routines to carry out multi-precision arithmetic on the computer, the processors used need to be of very high capabilities available on supercomputers. Using the Cray-2 supercomputer operated by the Numerical Aerodynamic Simulations (NAS) program at NASA Ames Research Center, David Bailey has implemented the Borwein algorithm to calculate π to 29,360,000 digits. 5. Conclusion As we can see both the ancient method and the modern method discussed here have their own merits and stand testimony to the creativity exhibited by the respective mathematicians in their times. While Archimedes was the first to propose an iterative method to compute the value of π using an iterative process way back in 250 B.C., the Borwein algorithm discovered a method to quadruple the number of digits to which π is computed from one iteration to the next, at phenomenally reduced processing times. The quest for improving the methods to unravel the mystery behind π will probably continue for a long time. Bibliography Beckmann Petr. A History of Pi. 3rd Ed. St. Martins Griffin. 1956. Print. Borwein Jonathan, Borwein Peter & Berggren Lennart. Pi: A Source Book. 3rd Ed. Springer Publication. New York. 2004. Web. 17 Nov 2009. Bucknall Julian. Calculating Pi. PCPlus Issue 228. Web. 17 Nov 2009. Heath Thomas. The Works of Archimedes. Dover Publications. 2002. Print. Sondow. Transcendental Number. Wolfram Mathworld. Web. 16 Nov 2009. Taylor Brian. 3.14>>History and Philosophy of Pi. Web. 16 Nov 2009. Read More