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Finding the Elusive Value of Π Juan Andrade 10/9/2016 Math 4388 Essay #2 Finding the elusive value of π Introduction Early mathematics mainly centered on geometry, due to the fact that the old world was filled with a variety of shapes for which numerical values had real life practical uses. For instance finding the area of a section of land, or finding the perimeter to which you want to build a fence for. When dealing with a shape that included a circle, finding the area or perimeter wasn’t as easy, and such a task was ideal for early mathematicians to tackle. Many attempts were made to find the value of π using various different techniques. While early attempts weren’t accurate enough, they did aid in finding the value of π that we use today. What is π, and why is it so important? A simple definition of π would be that it is a ratio of the circumference of a circle to twice of its radius, or its diameter. It allows us to calculate the perimeter and area of a circle by only knowing its radius, as well as similar calculations for more complex rounded shapes such as cones, cylinders, and spheres. The more exact value that was determined for π allowed for a more accurate representation of a circular shapes in the real world such as in cars, pipes and buildings. The advancement of the value of π and mathematics went hand in hand, and allowed for the more complex studies of limits, calculus and algorithms we study today. Earliest attempts to find π The first attempts to determine a value for π can be traced back to several thousands of years ago. Many civilizations knew that it was a ratio between the circumference and the Juan Andrade 10/9/2016 Math 4388 Essay #2 diameter of a circle, and their earliest estimates at around 3 were mostly determined through actual measurements or trial and error. The reason for the lack of precision in their estimated value could have been attributed to the slight discrepancies in the shapes that they were measuring. Furthermore, since the value was known to be a constant during this time, further trial and error attempts of circular objects should have led to a more accurate measurement for the value of π. Since this wasn’t the case many can only assume again that the objects that they used were slightly off from that of a perfect circle. Then again early civilizations didn’t know the value of π that we have today, and thus didn’t know how far off they were from finding an accurate value. Advanced civilizations got a more accurate estimate, the Egyptians got a value of 3.125 and the Babylonians a value of 3.16. Mostly a practical measurement of π sufficed during this time, until Archimedes tried a new method roughly 2200 years ago. Archimedes’ polygon method Archimedes’ attempt to determine the value of π involved the use of polygons inside and outside of an ideal circle. This method was different than that of previously used techniques because it was an iterative process, meaning the method could be repeated a number of times until achieving a desired precision. With polygons inside and outside of the boundary of a circle, Archimedes determined that the circumference of a circle could be calculated to be within the values of the perimeters of the polygons. For instance in his first attempt using trig and squares for his polygons, he calculated the perimeters to be 2.8 from 4*sin (45) = 2.8, and 4 from 4*tan (45) =1. Indicating that the circumference of a circle of radius 1 had to be between 2.8 and 4. Using 16 sided polygons for example would give us a perimeter of 16*sin (22.5/2) = 3.121 as the inside boundary, while 16*tan (22.5/2) =3.183 would be our outside boundary. Using 64 sided Juan Andrade 10/9/2016 Math 4388 Essay #2 polygons would give us a perimeter of 64*sin (5.625/2) = 3.140 as the inside boundary, while 64*tan (5.625/2) = 3.144 would be our outside boundary, and so on. As we can see from these previous examples, the more sides the polygons had the more accurate measurement of π was calculated (Figure 1). Since Archimedes didn’t have the convenience of our modern technology, he relied mostly on proofs and derivations when using his method. Such a time consuming process that it was, Archimedes calculated π using up to 96 sided polygons achieving an accuracy of 99.9%. A neat program allows us to play around with this method just by plugging in the number of sides of the polygons we want to use, and it gives us an output with an estimate of π and the percent accuracy (Figure 2). Figure 1. A simple graphic showing how the more sides the polygons along the boundary of the circle had, the more closely they resembled the circumference of the circle. Which is essentially how Archimedes’ method worked. Juan Andrade 10/9/2016 Math 4388 Essay #2 Figure 2. instacalc program for finding the value of π using Archimedes’ method. https://instacalc.com/2334 Other estimates for the value of π Using a similar method to that of Archimedes, Chinese mathematician Liu Hui accounted for the area of the polygons instead of the perimeter like Archimedes. With a formula of an area of a polygon of Pn = (1/2) n Sn, where n = # of sides, he was able to achieve an accuracy to 5 decimal places (Figure 3). Zhu Chongzhi further refined Liu Hui’s method and was able to achieve a value accurate to 7 decimal places, or 3.1415926. It took roughly a millennium from 500 AD to achieve a better estimate for π. Ludolph van Ceulen used the initial method introduced by Archimedes’, and a polygon of 15 x 2^31 sides to achieve an accuracy of up to 20 decimal places. As math got more complex so the accuracy in the measurement for π, and since the early 1700’s, most of the attempts were more accurate than the previous and didn’t involve the use of polygons. John Machin in 1706, established a formula for his approximation of π correct to 100 decimal places (Figure 4). Using Machin’s Formula in 1873, after the discovery that pi was irrational, Shanks continued the pursuit for a better π by achieving a value correct to 527 decimal places. Shanks attempt was one of the last made by a mathematician, since many figured it was pointless, and it was only after the invention of computers that we were able to get even more accurate values for π (Figure 5). Juan Andrade 10/9/2016 Math 4388 Essay #2 Figure 3. Liu Hui’s method to find a more accurate value for π. Figure 4. Machin’s formula for determining a value for π. Figure 5. Chronological achievements In the search for the value of π. Today’s uses of π Juan Andrade 10/9/2016 Math 4388 Essay #2 The search for the value of π could arguably be the most influential in the advancement of mathematics. Apart from still being used to find perimeter and area, π today has more complex uses in modern derivations, theorems, and calculus. Pi is used every day by architects and engineers in the construction of new bridges and buildings. Used to run power efficiently through power lines and water through underground pipes, as well as in the manufacturing of complex machines such as cars, ships, and planes. To put it simply “Anything with curvature has π” (O’Connor, 2011). If it wasn’t for all of those attempts to find π, we never would have had such innovations like those mentioned. Computers today even use π to determine their accuracy and processing speed to further improve our technology. Conclusion Everyone who worked in trying to find a value for π though history played a pivotal role in how we live and use π today. If no one had attempted to achieve a more accurate measurement after the initial 3 or 3.14, the math and world of today wouldn’t be as complex as it is. And through these achievements we now how the luxury of having buttons in our calculators with accurate depictions of the number π. Juan Andrade 10/9/2016 Math 4388 Essay #2 References “The values of pi through time”. Math forum. 1994-2016. Web 7 October 2016. < http://mathforum.org/isaac/problems/pi2.html> Azad, Kalid. “Prehistoric Calculus: Discovering Pi”. Better Explained. Web 6 October 2016. < https://betterexplained.com/articles/prehistoric-calculus-discovering-pi/> Huberty, Michael, Hayashi Ko, Chia Vang. “Historical Overview of Pi”. 6 July 1997. Web. 5 October 2016. < http://www.geom.uiuc.edu/~huberty/math5337/groupe/overview.html> O’Connor, JJ, Robertson EF. “A history of pi”. Pi through the ages. University of St Andrews, Scotland. 1 August 2011. Web 4 October 2016. < http://www-history.mcs.st- andrews.ac.uk/HistTopics/Pi_through_the_ages.html> O’Connor, JJ, Robertson EF. “Liu Hui” University of St Andrews, Scotland. 1 December 2013. Web 6 October 2016. < http://www-history.mcs.st-andrews.ac.uk/Biographies/Liu_Hui.html> O’Connor, JJ, Robertson EF. “Ludolph Van Ceulen” University of St Andrews, Scotland. 1 April 2009. Web 6 October 2016.< http://www-history.mcs.st-andrews.ac.uk/Biographies/Van_Ceulen.html> Turner, Lawrence. “Arctan Formulae for Computing π”. Mathematics. 10 July 2016. Web 6 October 2016.
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