A Brief History of the Approximation of π

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1 A Brief History of the Approximation of π Communication Presentation Nicole Clizzie Orion Thompson-Vogel April 3, 2017

2 Outline Archimedes Archimedes uses Euclid s Theorem Francois Viete First Infinite Series for π Gottfried Leibniz, James Gregory, and Nilakantha Leibniz-Gregory Series John Machin Machin Series Srinivasa Ramanujan Ramanujan s Approximation of π

3 Outline Archimedes Archimedes uses Euclid s Theorem Francois Viete First Infinite Series for π Gottfried Leibniz, James Gregory, and Nilakantha Leibniz-Gregory Series John Machin Machin Series Srinivasa Ramanujan Ramanujan s Approximation of π

4 Archimedes (287 BC-212 BC) Archimedes uses Euclid s Theorem Theorem Euclid s Theorem If a straight line bisects an angle of a triangle and cuts the base then the resulting segments of the base have the same ratio as the remaining sides of the triangle. Let s now prove this theorem.

5 Archimedes (287 BC-212 BC) Archimedes uses Euclid s Theorem Proof. If we have AOC and OD bisects AOC, then OC OA = CD DA. So the first step is to draw AOC and subsequently draw in the angle bisector OD. Afterwards, we may add a point E such that it forms a line EC, which is parallel to OD, and a line AE, where AE is an extension of AO. Here we can reason a few things:

6 Archimedes (287 BC-212 BC) Archimedes uses Euclid s Theorem Proof. m DOA=m CEA since they re corresponding angles m OCE+m CEO+m EOC=180 as these form a triangle m AOD+m DOC+m EOC=180 since they re supplementary m DOC+m AOD=m OCE+m CEO with substitution COE is isosceles by definition and using AA similarity theorem, DOA is similar to CEA so

7 Archimedes (287 BC-212 BC) Archimedes uses Euclid s Theorem Proof. CA AE = DA AO CA DA = AE AO AO+OE AO = CD+DA DA OE AO = CD DA CO AO = CD DA and since CA=CD+DA and AE=AO+OE

8 Outline Archimedes Archimedes uses Euclid s Theorem Francois Viete First Infinite Series for π Gottfried Leibniz, James Gregory, and Nilakantha Leibniz-Gregory Series John Machin Machin Series Srinivasa Ramanujan Ramanujan s Approximation of π

9 Archimedes (287 BC-212 BC) Let O be the center and AB be the diameter while AC is tangent at A. Let m AOC be 30 degrees or π 6 radians. So note CF = 2 CA where CF is a side of a circumscribed hexagon. Also note that where n is the number of sides, n AC Perimeter of the polygon OA Diameter = π. Now let s bisect m AOC, and using Euclid s theorem, CD DA = CO OA we see CD+DA DA = CO+OA OA = CA DA CO+OA CA = OA DA and so (i) CO CA + OA CA = OA AD.

10 Archimedes (287 BC-212 BC) Let O be the center and AB be the diameter while AC is tangent at A. Let m AOC be 30 degrees or π 6 radians. So note CF = 2 CA where CF is a side of a circumscribed hexagon. Also note that where n is the number of sides, n AC Perimeter of the polygon OA Diameter = π. Now let s bisect m AOC, and using Euclid s theorem, CD DA = CO OA we see CD+DA DA = CO+OA OA = CA DA CO+OA CA = OA DA and so (i) CO CA + OA CA = OA AD.

11 Archimedes (287 BC-212 BC) Now using Pythagorean Theorem: OA 2 + AD 2 = OD 2 OA2 + 1 = AD OD2 2 DA 2 OA (ii) = OD AD 2 DA. Archimedes then approximates 3 so OA AC = 3 since OA AC = cot( π 6 ) so OA AC >

12 Archimedes (287 BC-212 BC) Now using Pythagorean Theorem: OA 2 + AD 2 = OD 2 OA2 + 1 = AD OD2 2 DA 2 OA (ii) = OD AD 2 DA. Archimedes then approximates 3 so OA AC = 3 since OA AC = cot( π 6 ) so OA AC > also note that OC CA = 2 1 = as for us this is 1 sin( π 6 (i) we have < OA AD < OA AD ) and using

13 Archimedes (287 BC-212 BC) Let s find OD DA using (ii). So since we know something about OA AD we can plug this in knowing it is less than the actual value so we get OA2 + 1 > AD and OA > AD which Archimedes reduces to OD AD =

14 Archimedes (287 BC-212 BC) And now the next step is to bisect AOD by OE, and using the same approach as before, we have OA and so OA AE > = AE = OD DA + OA DA 153 and so this gives us a 12-gon s approximation. [4] We can continue this until we have a 96-gon, which is what Archimedes did, where he achieved an upper-bound of and a lower bound of [8]

15 Outline Archimedes Archimedes uses Euclid s Theorem Francois Viete First Infinite Series for π Gottfried Leibniz, James Gregory, and Nilakantha Leibniz-Gregory Series John Machin Machin Series Srinivasa Ramanujan Ramanujan s Approximation of π

16 Francois Viete ( ) First Infinite Series for π Lawyer in France who worked for King Henry III and Henry IV Dabbled in quite a bit of mathematics, as in 1593 Adriaan van Roomen, a mathematician from the Netherlands, challenged french mathematicians with a problem posed about a 45 degree polynomial. An ambassador from the Netherlands commented on the poor quality of french mathematicians as none could solve this problem, however, this prompted King Henry to present the problem to Viete, who found a solution on the first day and 22 more the next day. First infinite expansion of π 2 π = A variation of Archimedes but instead of using polygons perimeters he used the areas. [5]

17 Outline Archimedes Archimedes uses Euclid s Theorem Francois Viete First Infinite Series for π Gottfried Leibniz, James Gregory, and Nilakantha Leibniz-Gregory Series John Machin Machin Series Srinivasa Ramanujan Ramanujan s Approximation of π

18 Gottfried Leibniz ( ), James Gregory ( ), and Nilakantha Liebniz-Gregory Series Liebniz-Gregory Series π = 4( ) So we may quickly prove this case using the Taylor series. The Taylor series is the following: 1 1 y = 1 + y + y and substituting in y = x 2 we get 1 1+x 2 = 1 x 2 + x 4... and note that the left hand side is the same as d dx tan(x) 1 and so if we integrate both sides we have tan(x) 1 = x x3 3 + x and if we plugin x=1, and multiply the right side by 4, we have our result. [10]

19 Outline Archimedes Archimedes uses Euclid s Theorem Francois Viete First Infinite Series for π Gottfried Leibniz, James Gregory, and Nilakantha Leibniz-Gregory Series John Machin Machin Series Srinivasa Ramanujan Ramanujan s Approximation of π

20 John Machin ( ) Machin Series π 4 = artan 1 5 arctan so we can derive this as follows: let tan(α)=1/5 where alpha is some angle and now we see using the double angle formula tan(2α)= 2 tan(α) 1 tan 2 (α) = = 5 12 and repeating again tan(4α)= 2 tan(2α) 1 tan 2 (2α) = 5 6 = Now note that this differs from 1 by only a tiny amount, and so we can examine the following: tan(4α π 4 ) = tan(4α) tan( π 4 tan(4α)+tan( π 4 taking the arctan of both sides gives 4α- pi 4 = arctan = tan(4α) 1 tan(4α+1) = = and which simplified to the result above. [3]

21 Outline Archimedes Archimedes uses Euclid s Theorem Francois Viete First Infinite Series for π Gottfried Leibniz, James Gregory, and Nilakantha Leibniz-Gregory Series John Machin Machin Series Srinivasa Ramanujan Ramanujan s Approximation of π

22 Srinivasa Ramanujan ( ) Ramanujan s Approximation of π Ben Lynn from Stanford University stated Its my favourite formula for pi. I have no idea how it works. 1 π = 8 (4n)! 9801 n= n+1103 (n!) n The formula involved modular equations. [2]

23 Summary A good approximation of π was first done by Archimedes, and all that have followed suit have involved infinite series. Some approximations are better than others due to their convergence rate. The mysterious Ramanujan crafted the essence of the best π approximation formula to date.

24 Bibliography I Ben, Lynn. The Gregory-Leibniz Series, Standford. Web 2 Apr Ben, Lynn. Ramanujans Formula for Pi, Standford. Web 2 Apr Derivation of the Machin Formula. Arctan Formulae for Computing Pi. Southwestern Adventist University. Web. 2 Apr Han, Kyutae Paul.Pi and Archimedes Polygon Method, Dartmouth College. Web 28 Apr Hartshorne, Robin. Francois Viete-Life. Mathematicans. Web 2 Apr O Connor, J. J., and E. F. Robertson, John Machin, Historical Topics, School of Mathematics and Statistics University of St. Andrews, April Web 2 Apr

25 Bibliography II O Connor, J. J., and E. F. Robertson, Gottfried Wilhelm von Liebniz, Historical Topics, School of Mathematics and Statistics University of St. Andrews, Oct Web 2 Apr O Connor, J. J., and E. F. Robertson, Archimedes of Syracuse, Historical Topics, School of Mathematics and Statistics University of St. Andrews, Jan Web 2 Apr O Connor, J. J., and E. F. Robertson, Srinivasa Aiyangar Ramanujan, Historical Topics, School of Mathematics and Statistics University of St. Andrews, Jun Web 2 Apr Roy, Ranjan. The Discovery of the Series Formula of Pi by Leibniz, Gregory, and Nilakatha. Mathematics Magazine Dec. 1990: Web 2 Apr

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