A Brief History of the Approximation of - Communication...

A Brief History of the Approximation of πCommunication Presentation

Nicole Clizzie Orion Thompson-Vogel

April 3, 2017

Outline

ArchimedesArchimedes uses Euclid’s TheoremUpper and Lower Bounds of a Polygon

Francois VieteFirst Infinite Series for π

Gottfried Leibniz, James Gregory, and NilakanthaLeibniz-Gregory Series

John MachinMachin Series

Srinivasa RamanujanRamanujan’s Approximation of π

Archimedes (287 BC-212 BC)Archimedes uses Euclid’s Theorem

I TheoremEuclid’s Theorem If a straight line bisects an angle of a triangleand cuts the base then the resulting segments of the base have thesame ratio as the remaining sides of the triangle.

I Let’s now prove this theorem.


I Proof.If we have 4 AOC and OD bisects ∠ AOC, then OC

OA =CDDA . So the

first step is to draw 4 AOC and subsequently draw in the anglebisector OD. Afterwards, we may add a point E such that it formsa line EC, which is parallel to OD, and a line AE, where AE is anextension of AO. Here we can reason a few things:


I Proof.m∠DOA=m∠CEA since they’re corresponding anglesm∠OCE+m∠CEO+m∠EOC=180 as these form a trianglem∠AOD+m∠DOC+m∠EOC=180 since they’re supplementarym∠DOC+m∠AOD=m∠OCE+m∠CEO with substitution4 COE is isosceles by definitionand using AA similarity theorem, DOA is similar to 4 CEA so


I Proof.CAAE = DA

AO ⇒CADA = AE

AO and since CA=CD+DA and AE=AO+OEAO+OE

AO = CD+DADA

OEAO = CD

DACOAO = CD

DA

Outline






Archimedes (287 BC-212 BC)Upper and Lower Bounds of a Polygon

I

Let O be the center and AB be the diameter while AC istangent at A. Let m∠AOC be 30 degrees or π

6 radians. Sonote CF = 2 ∗ CA where CF is a side of a circumscribedhexagon. Also note that where n is the number of sides,n ∗ AC

OAPerimeter of the polygon

Diameter = π. Now let’s bisect m∠AOC,

and using Euclid’s theorem, CDDA = CO

OA we seeCD+DA

DA = CO+OAOA = CA

DA ⇒CO+OA

CA = OADA and so

(i) COCA + OA

CA = OAAD .


I

Now using Pythagorean Theorem: OA2 + AD2 = OD2

⇒ OA2

AD2 + 1 = OD2

DA2

(ii)√

OA2

AD2 + 1 = ODDA . Archimedes then approximates

√3 so

OAAC =

√3 since OA

AC = cot(π6 ) so OAAC > 265

153

also note that OCCA = 2

1=306153 as for us this is 1

sin(π6) and using

(i) we have306153 + 265

153 <OAAD ⇒

571153 <

OAAD


I

Let’s find ODDA using (ii). So since we know something about

OAAD we can plug this in knowing it is less than the actual value

so we get OA2

AD2 + 1 > 34945023409 and

√OA2

AD2 + 1 >√

34945023409 which

Archimedes reduces to ODAD =

591 18

153 .


I

And now the next step is to bisect 4AOD by OE, and usingthe same approach as before, we have OA

AE = ODDA + OA

DA and so

OAAE >

591 18

153 + 571153 =

1162 18

153 and so this gives us a 12-gon’sapproximation. [4] We can continue this until we have a96-gon, which is what Archimedes did, where he achieved anupper-bound of 31

7 and a lower bound of 31071 . [8]

Outline






Francois Viete (1540-1603)First Infinite Series for π

I Lawyer in France who worked for King Henry III and Henry IV

I Dabbled in quite a bit of mathematics, as in 1593 Adriaan vanRoomen, a mathematician from the Netherlands, challengedfrench mathematicians with a problem posed about a 45degree polynomial. An ambassador from the Netherlandscommented on the poor quality of french mathematicians asnone could solve this problem, however, this prompted KingHenry to present the problem to Viete, who found a solutionon the first day and 22 more the next day.

I First infinite expansion of π

2π =

√22 ·√

2+√2

2 ·√

2+√

2+√2

2 · ··A variation of Archimedes but instead of using polygons’perimeters he used the areas. [5]

Outline






Gottfried Leibniz (1646-1716), James Gregory(1638-1675), and NilakanthaLiebniz-Gregory Series

Liebniz-Gregory Series

π = 4(1− 13 + 1

5 −17 + 1

9 − · · ·)So we may quickly prove this case using the Taylor series. TheTaylor series is the following:1

1−y = 1 + y + y2 + ... and substituting in y = −x2 we get1

1+x2= 1− x2 + x4 − ... and note that the left hand side is the

same as ddx tan(x)−1 and so if we integrate both sides we have

tan(x)−1 = x − x3

3 + x5

5 − ... and if we plugin x=1, and multiplythe right side by 4, we have our result. [10]

Outline






John Machin (1690-1751)Machin Series

I π4 = artan 1

5 − arctan 1239

so we can derive this as follows:let tan(α)=1/5 where alpha is some angleand now we see using the double angle formula

tan(2α)= 2∗tan(α)1−tan2(α)=

25

1− 125

= 512

and repeating again

tan(4α)= 2∗tan(2α)1−tan2(2α)=

56

1− 25144

= 120119 .

Now note that this differs from 1 by only a tiny amount, andso we can examine the following:

tan(4α− π4 ) =

tan(4α)−tan(π4

tan(4α)+tan(π4

= tan(4α)−1tan(4α+1) =

120119−1

120119

+1= 1

239 and

taking the arctan of both sides gives4α-pi4 = arctan 1

239 which simplified to the result above. [3]

Outline






Srinivasa Ramanujan (1887-1920)Ramanujan’s Approximation of π

I Ben Lynn from Stanford University stated ”Its my favouriteformula for pi. I have no idea how it works.”1π =

√8

9801

∑∞n=0

(4n)!(n!)4× 26390n+1103

3964n

I The formula involved modular equations. [2]

Summary

I A good approximation of π was first done by Archimedes, andall that have followed suit have involved infinite series.

I Some approximations are better than others due to theirconvergence rate.

I The mysterious Ramanujan crafted the essence of the best πapproximation formula to date.

Bibliography I

Ben, Lynn. The Gregory-Leibniz Series, Standford. Web 2 Apr.2017.

Ben, Lynn. Ramanujans Formula for Pi, Standford. Web 2Apr. 2017.

Derivation of the Machin Formula. Arctan Formulae forComputing Pi. Southwestern Adventist University. Web. 2 Apr.2017.

Han, Kyutae Paul.Pi and Archimedes Polygon Method,Dartmouth College. Web 28 Apr. 2017.

Hartshorne, Robin. Francois Viete-Life. Mathematicans. Web 2Apr. 2017.

O’Connor, J. J., and E. F. Robertson, John Machin, HistoricalTopics, School of Mathematics and Statistics University of St.Andrews, April 2003. Web 2 Apr. 2017.

Bibliography II

O’Connor, J. J., and E. F. Robertson, Gottfried Wilhelm vonLiebniz, Historical Topics, School of Mathematics andStatistics University of St. Andrews, Oct 1998. Web 2 Apr.2017.

O’Connor, J. J., and E. F. Robertson, Archimedes of Syracuse,Historical Topics, School of Mathematics and StatisticsUniversity of St. Andrews, Jan 1999. Web 2 Apr. 2017.

O’Connor, J. J., and E. F. Robertson, Srinivasa AiyangarRamanujan, Historical Topics, School of Mathematics andStatistics University of St. Andrews, Jun 1998. Web 2 Apr.2017.

Roy, Ranjan. The Discovery of the Series Formula of Pi byLeibniz, Gregory, and Nilakatha. Mathematics Magazine Dec.1990: 291-306. Web 2 Apr. 2017.

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