SPECIAL SECTION Civilisation Dated 1900-1600 BC Has a Circle's Circumference Instead of the Statement That Implicates a Value of Area

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SPECIAL SECTION civilisation dated 1900-1600 BC has a circle's circumference instead of the statement that implicates a value of area. This algorithm was devised around 3.1250 to pi. In the Egyptian civilisation 250 BC and dominated for over 1000 (around 1650 BC), the area of a circle years, as a result of which pi is was calculated by using an approximate sometimes referred as ‘Archimedes value of 3.1605 as pi. In Egypt, the constant’. He developed a polygon Rhind Papyrus, dates around 1650 BC, approach to approximate the value of pi but copied from a document dated 1850 and found the upper and lower bounds BC mentions the formula for the area of of pi by inscribing and circumscribing a a circle that treats pi as 3.1605. In India circle in a hexagon, and successively around 600 BC, some Sanskrit texts doubling the number of sides. He did treat pi as equal to 3.088. this until he obtained a 96-sided polygon. By calculations involving the When Greeks took up the problem, they perimeters of these polygons, he took two revolutionary steps to find pi. obtained the bounds of pi and proved Antiphon and Bryson of Heraclea came that 3.1408 < pi< 3.1429. Around 150 up with the idea of drawing a polygon AD, Greek-Roman scientist Ptolemy inside a circle, finding its area, and gave a value of 3.1416 in his Almagest, doubling the sides repeatedly. Later, which he may have obtained from Bryson also calculated the area of Archimedes or from Apollonius of polygons circumscribing the circle. Perga. Using this algorithm, Perhaps this was first time that a mathematicians reached 39 digits of pi mathematical result for pi was in 1630, a record unbroken until 1699 determined by using upper and lower when the infinite series approach was bounds. The first man who really made devised. a serious attempt to calculate the value of pi was a Greek scientist Archimedes In China, in around 265 AD, Liu Hui, a of Syracuse. Archimedes carried Wei kingdom mathematician created a forward the work of Antiphon and polygon-based iterative algorithm and Bryson. However, his approach was implemented it with a 3072-sided slightly different from Antiphon and polygon to obtain a value of 3.1416. He Bryson. Archimedes focused on the later invented a faster method and polygon's perimeters as opposed to their obtained a value of 3.14 with a 96-sided areas, so that he approximated the polygon, using the fact that differences

New Horizons, Vol. 2 December 2016 SPECIAL SECTION in the area of successive polygons form European mathematicians used infinite a geometric series with a factor of 4. series to compute pi with greater Another mathematician ZuChongzhi accuracy. The first infinite sequence was applied Liu’s algorithm to a 12288- discovered by French mathematician sided polygon and calculated the correct Francois Viete. He used Archimedean value of its first seven decimal digits. method. Francois started with two This value remained the best hexagons and then doubling the number approximation for the next 800 years. In of sides sixteen times, and finally the 5th century, TsuCh'ung-chih and his finishing with 393,216 sides. His final son TsuKeng-chih calculated the value result was that 3.1415926535 < pi < of pi ranging between 3.1415926 and 3.1415926537. Adrianus Romanus, in 3.1415927 by inscribing polygon with 1593, computed pi with the accuracy to as many as 24,576 sides. In the same 15 digits after the decimal by using a period, the Hindu mathematician circumscribed polygon with 230 sides. Aryabhata calculated better After few years, Ludolph Van Ceulen, a approximation 62,832/20,000 = 3.1416 German presented 20 digits, using the (as against Archimedes’ 22/7 which was Archimedean method with polygons frequently used), but this value did not with over 500 million sides. Van Ceulen get any noticeable attention. Another spent a great part of his life hunting for Indian mathematician, Brahmagupta, pi, and he had computed pi up to 35 took a novel approach by calculating the digits after decimal. To recognize his perimeters of inscribed polygons with work, the digits were written into his 12, 24, 48, and 96. He observed that as tombstone in St. Peter's Churchyard in the polygons approached the circle, the Leyden. Here, it is important to mention perimeter, and therefore pi, would that up to this time, there was no symbol approach the square root of 10 [= to denote the ratio of a circle's 3.162...]. In 9th century, mathematics circumference to its diameter. This and science prospered in the Arab changed in 1647 when William cultures. It is stated that, in some texts, Oughtred published Clavis that the Arabian mathematician, Mathematicae and used (/(to denote the Mohammed bin Musa al'Khwarizmi, ratio. It was not accepted immediately attempted to calculate pi approximately but in 1737 when Leonhard Euler began 21/7. During the 16th and 17th centuries, using the symbol for pi; then it was

New Horizons, Vol. 2 December 2016 SPECIAL SECTION quickly accepted. In 1650, John Wallis William Shanks used the formula to used a very complicated method to find calculate 707 places of pi. another formula for pi. He approximated Many years later, it was discovered that the area of a quarter circle using somewhere along the line, Shanks had infinitely small rectangles, and arrived left two terms as a result only the first at the formula for 4/pi which is 527 digits were correct. Machin-like simplified to pi/2. In 1672, James variants were subsequently created by Gregory wrote about a formula that can others who created records for 250 years be used to calculate the angle given the and more. Daniel Ferguson achieved a tangent for angles up to 45pi. After ten 620-digit approximation in 1946, which years, Gottfried Leibnitz pointed out was the best possible approximation that since tan pi/4 = 1, the formula could without the use of a calculating device. be used to find pi. Thus, one of the By 1750, the number pi had been most famous formulas for calculating pi expressed by infinite series, its value was realized: (/4 = 1 - 1/3 + 1/5 - 1/7 + had been computed [to over 100 digits] 1/9.... This elegant formula is one of the and it had been given its present symbol. simplest ever discovered to calculate pi, All these efforts, however, had not but it was also useless; 300 terms of the contributed to the solution of the ancient series are required to get only 2 decimal problem of the quadrature of the circle. places, and 10,000 terms are required The first step was taken by the Swiss for 4 decimal places. To compute 100 mathematician Johann Heinrich digits, one has to calculate more terms Lambert when he proved the than there are particles in the universe. irrationality of pi first in 1761 and then Better approximations of PI in more detail in 1767. His argument In 1706, John Machin, a professor of was, in its simplest form, that if x is a astronomy in London, took an initiative rational number, then tan x cannot be to calculate pi with his new formula, and rational; since tan pi/4 = 1, pi/4 cannot computed 100 places by hand. Over the be rational, and therefore pi is irrational. next 150 year, several men used the Some people felt that his proof was not same formula to find more and more rigorous enough, but in 1794, Adrien digits. In 1873, an Englishman named Marie Legendre gave another proof that satisfied everyone. Furthermore,

New Horizons, Vol. 2 December 2016 SPECIAL SECTION

Legendre also gave the first proof that in 1949, another breakthrough emerged, square root of 2 is irrational. In 1882, but it was not mathematical in nature; it Ferdinand von Lindemann proved the was the speed with which the transcendence of pi. Since this means calculations could be done. The ENIAC that pi is not a solution of any algebraic (Electronic Numerical Integrator and equation, it lay to rest the uncertainty Computer) was finally completed and about squaring the circle. Finally, after functional, and a group of literally thousands and thousands of mathematicians headed by John von lifetimes of mental toil and strain, Neumann fed in punch cards and let the mathematicians finally had an absolute gigantic machine calculate 2037 digits answer that the circle could not be in just seventy hours; whereas it took squared. Nonetheless, there are still Shanks several years to come up with some amateur mathematicians today his 707 digits, and Ferguson needed who do not understand the significance about one year to get 808 digits. With of this result, and look for techniques to the invention of the electronic computer, square the circle. John Wrench and Daniel Shanks found 100,000 digits in 1961, and the one- PI in computer Era million-mark was surpassed in 1973. In The computer era further saw 1976, Eugene Salamin developed an developments in the calculation of more algorithm that doubles the number of number of digits of pi. The discovery of correct digits with each iteration. After iterative algorithms independent of the development of this algorithm, the infinite series, and fast multiplication number of digits of pi increased algorithms led to further increase in enormously. After 1980, iterative these developments. In 1945, D. F. algorithms that were faster than infinite Ferguson discovered the error in series algorithms were used, since William Shanks' calculation from the iterative algorithms multiply the number 528th digit onward. Two years later of digits at each step. Over the past Ferguson presented his results after an twenty years, Yoshiaki Tamura, entire year of calculations, which Yasumasa Kanada, Jonathan, Peter resulted in 808 digits of pi. One and a Borwein, David and Gregory half years later, Levi Smith and John Chudnovsky have done tremendous Wrench hit the 1000-digit-mark. Finally, work for pi. In fact, Kanada and Tamura

New Horizons, Vol. 2 December 2016 SPECIAL SECTION worked together on many pi projects, was from pi? We don't really look for and led the way throughout the 1980s. patterns; we look for rules But the Chudnovskys broke the one- Euler established a connection between billion-barrier in August 1989. In 1997, pi and the prime numbers by solving the Kanada and Takahashi, surprisingly, Basel problem, that later helped in the calculated 51.5 billion (3(234) digits in study of the Riemann zeta function. The just over 29 hours, at an average rate of record for calculating pi, as of 2010 is to nearly 500,000 digits per second! The 5 trillion digits. The fraction 22/7 which current record, set in 1999 by Kanada is generally used as an approximation of and Takahashi, is 68,719,470,000 digits. pi is accurate to 0.04025%. The digits of We hope many more results. However, pi have no specific pattern, and are certainly, the continued calculations are statistically random in occurrence. They unnecessary. Just thirty-nine decimal have even passed the tests of normality, places would be sufficient to compute which implies that all possible the circumference of a circle sequences of digits appear equally often. surrounding the known universe to Despite this, there exist some sequences within the radius of a hydrogen atom. which may appear non-random, like the Surely, there is no conceivable need for Feynman Point, which is a sequence of billions of digits. Presently, the only six 9s starting from the 762nd decimal application for all those digits is to test representation of pi. computers and computer chips for bugs. Various approximations in different But digits aren't really what number systems exist that are widely mathematicians are looking for used. Some approximate fractions used anymore. As the Chudnovsky brothers are 22/7, 333/106, etc. In decimal once said: "We are looking for the system, it is generally written as 3.1415. appearance of some rules that will The binary approximation is written as distinguish the digits of pi from other 11.0010. numbers. If you see a Russian sentence The ubiquity of pi in formulae has led to that extends for a whole page, with it being represented in popular culture hardly a comma, it is definitely Tolstoy. more than any other mathematical If someone gave you a million digits constant. Several songs and movies have from somewhere in pi, could you tell it lyrics and dialogues incorporating pi. A special day called Pi day is also

New Horizons, Vol. 2 December 2016 SPECIAL SECTION celebrated on March 14th because of its Pi Approximation day is also observed date i.e. 3/14, which are first three on July 22nd, i.e. 22/7 format because it significant digits in approximation of pi. is the common approximation of pi.