History Of Maths: Pi

The discovery and refinement of Pi and its properties has been a continual process from Ancient civilisations through to modern times. The process of identifying the mathematical properties of Pi has involved contributions and ideas from many mathematicians both past and present. The methods used to develop and apply Pi have played a key role in the development of new ways of thinking and approaching mathematical problems. I will firstly look at the discovery of Pi in ancient civilisations followed by the development over time through to the modern era. Pi can be defined as a mathematical constant with the value often used to two decimal places as 3.14. It is also standard convention for Pi to be abbreviated by the Greek letter?. The value of Pi defines the ratio between the diameter and circumference of a circle as follows; Pi Circumference Diameter. The value of Pi is particularly useful in calculating the circumference and area of circles given their diameter. The first approximation and use of Pi can be traced back to Ancient Egypt in around 1650 BC. The Egyptians had an extensive understanding of mathematical problems and their practical application. This was highlighted by the discovery of the Rhind Papyrus as the earliest known reference to Pi and other complex mathematical problems. The Egyptian estimate of Pi is derived as Problem 50 in the Rhind Papyrus . The problem aims to calculate the area of a circular field of diameter of 9 Khet (unit of length measurement). The solution can be broken down into two stages. The first step in the solution is to calculate 1/9th of the diameter. This yields 1 with remainder 8 (9 Khet × 1/9=1). The second step multiplies the remainder 8 by 8 to give 64 Setjat (unit of square measurement). Using the exact value of Pi to solve the same problem gives a solution of 63.62 Setjat showing that the Egyptian estimate of Pi was only 0.6% of its true value . The Egyptians applied the mathematical concept of Pi in the construction of Pyramids. The knowledge of Pi helped the Egyptians construct pyramids to a high degree of accuracy despite limited engineering technology. The use of Pi was particularly important as it was used to calculate the angles of the pyramid walls to ensure they didn`t collapse. Rouse 1960 (page 5) also highlights that the development of Pi was "out of necessity of surveying" land. The move towards permanent settlements as opposed to nomadic lifestyles also meant that mathematical and geometry skills were needed for basic engineering projects such as the provision of accommodation and irrigation. The use of Pi was needed to calculate the allocation of land and also potential yields from each unit of land. Evidence of approximations of Pi was also found in Old Babylonia in around 1800 BC. During this period mathematics was recorded on clay tablets as opposed to papyrus scrolls. In Ancient Greece 240 BC a Greek mathematician Archimedes made further developments in circle geometry and also estimated a ratio for Pi known as the Archimedean algorithm. Due to his significant contribution Pi is also referred to as Archimedes Constant. Archimedes method involved calculating the perimeter of a hexagon within a circle. The number of sides was then doubled and the perimeter recorded. This process was repeated up to a polygon with 96 sides and yielded a more accurate estimate of Pi compared to the Egyptian and Babylonian calculations . Relative to the known value of Pi Archimedes approximation was only 0.0002 away from the true value. The method was accurate as Archimedes recognised the fact that as the number of sides of the polygon tends towards infinity the perimeter of the polygon tends to the circumference of the circle. In general terms a circle can be considered as a polygon with an infinite number of sides. Archimedes was also the first mathematician to recognise Pi could not be calculated as an exact number but was an infinite number between two estimated values 223 71 22 7 . Research into Pi also prospered in both India and China. Liu Hui was able to simplify Archimedes calculation by using a geometric series to work out the perimeter of polygons as the number of sides increased. Using this Liu Hui was able to derive the value of 3.14 for Pi through the use of a 96 side regular polygon in his commentary of the `The Nine Chapters on the Mathematical Art. In India Aryabhata also estimated the value of Pi to 3.1416. These developments were followed by a period of little progress until the renaissance period in Europe. The interest in Pi was stimulated by factors such as the growing role of mathematics in navigation, the introduction of Arabic numerals and the introduction of the decimal system. The discovery of calculus and the concept of infinite series led to a more accurate figure of Pi being discovered. In the year 1400 Madhava of Sangamagrama found an infinite sequence of numbers that converged to... leading to an 11 decimal place value of Pi. In the 1700s further developments came from Sir Isaac Newton, Leibniz and Euler whom developed new more efficient methods of calculating Pi. Pi was first proven to be irrational by Johann Lambert in a paper presented to the Berlin Academy in 1768 . This implies that Pi cannot be written as a fraction in the form A/B ; where A, B are integers and B is not equal to 0. The proof states that if x is non zero and rational; then neither x nor tan(x) can be rational. It follows that tan .. which implies that .. and Pi must be irrational. Some mathematicians have argued that Lamberts proof was incomplete and it was Legendre who devised the first complete proof of irrationality. The development of Pi has been a gradual process with the true value of Pi and its properties having been discovered across many years, cultures and mathematicians. In the modern era supercomputers and the use of complex algorithms have allowed the calculation for Pi to be completed to over a trillion decimal places . However, research into Pi continues with questions such as whether Pi is a normal number yet to be answered. Bibliography Books: 1. Clagett, M., (1999). Ancient Egyptian Science, A Source Book. Volume Three: Ancient Egyptian Mathematics. Philadelphia, Volume 232. American Philosophical Society. 2. Ball-Rouse, W., (1960). A Short Account of the History of Mathematics. New York, Dover Publications. 3. Berggren L, Borwein J and Borwein P.,(2004). Pi: A Source Book. New York, Third Edition. Springer Verlag-Publishers. Internet Sources: O`Connor and Robertson,. (2011). The MAC Tutor History of Mathematics Archive, University of St Andrews, Scotland. history.mcs.st-and.ac.uk Biography Index: Archimedes of Syracuse, Johann Lambert, Liu Hui History Topics Index: A Chronology of Pi Palmer, J,. (2010). Pi calculated to `record number` of digits. BBC Science and Technology news.bbc.co.ukslash1hi844225 5.stm A, Solli,. (2012). A Chronological History of Pi with Developmental Activities in Problem Solving. Yale-New Haven Teachers Institute. www.cis.yale.eduynhticurriculumunits19807800711xtml Accessed ORACLE Education Foundation Think Quest Library. `Your piece of the ?` library.thinkquest.orgC0110195historyhistory.html Complete website addresses of above references are available upon request