The Calculation of Paul Butterfoss
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Paul Butterfoss Zachary Tseng Math 251 – Extra Credit Paper 3.14159265358979323846264338327950288419716939937510582097494459230781640628620899862803482534211706798214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196442881097566593344612847564823378678316527120190914564856692346034861045432664821339360726024914127372458700660631558817488152092096282925409171536436789259036001133053054882046652138414695194151160943305727036575959195309218611738193261179310511854807446237996274956735188575272489122793818301194912983367336244065664308602139494639522473719070217986094370277053921717629317675238467481846766940513200056812714526356082778577134275778960917363717872146844090122495343014654958537105079227968925892354201995611212902196086403441815981362977477130996051870721134999999837297804995105973173281609631859502445945534690830264252230825334468503526193118817101000313783875288658753320838142061717766914730359825349042875546873115956286388235378759375195778185778053217122680661300192787661119590921642019893809525720106548586327886593615338182796823030195203530185296899577362259941389124972177528347913151557485724245415069595082953311686172785588907509838175463746493931925506040092770167113900984882401285836160356370766010471018194295559619894676783744944825537977472684710404753464620804668425906949129331367702898915210475216205696602405803815019351125338243003558764024749647326391419927260426992279678235478163600934172164121992458631503028618297455570674983850549458858692699569092721079750930295532116534498720275596023648066549911988183479775356636980742654252786255181841757467289097777279380008164706001614524919217321721477235014144197356854816136115735255213347574184946843852332390739414333454776241686251898356948556209921922218427255025425688767179049460165346680498862723279178608578438382796797668145410095388378636095068006422512520511739298489608412848862694560424196528502221066118630674427862203919494504712371378696095636437191728746776465757396241389086583264599581339047802759009946576407895126946839835259570982582262052248940772671947826848260147699090264013639443745530506820349625245174939965143142980919065925093722169646151570985838741059788595977297549893016175392846813826868386894277415599185592524595395943104997252468084598727364469584865383673622262609912460805124388439045124413654976278079771569143599770012961608944169486855584840635342207222582848864815845602850601684273945226746767889525213852254995466672782398645659611635 Paul Butterfoss Zachary Tseng Math 251
Pi () is probably the most well known mathematical symbol. It deals with everything from trigonometric functions to three-dimensional spheres. One of its most interesting characteristics is that it is a non-repeating decimal, a totally random series of numbers with no apparent pattern. Then how is pi calculated? That is what I plan to discus in the following pages. One method for calculating pi involves the MacLaurin series of the inverse tangent function (1). tan-1(x) = The MacLaurin series for a function (3) is found by taking the Taylor series (2) and evaluating for the special case a=0.  (x) = 
= (x) =  = (0) +  +  ……. (3)
When x=1 the MacLaurin series for tan-1(x) is equal to the series expansion for /4. This is called the Leibniz series (4). (This formula is attributed to Leibniz, although it now seems that James Gregory (1638-1675) first discovered it). tan-1(1) = To compute the number of terms of the Leibniz series needed to reach /4 to a certain accuracy, we must use the Alternating Series Estimation Theorem (5). If s = (-1)n-1bn is the sum of an alternating series that satisfies (a) 0 bn+1 bn and (b)  
then |Rn| = |s – sn| bn+1 (5) In order to reach 6-digit accuracy, bn+1 must be 0.000001. Therefore, the greatest possible value for bn+1 is 1/1000001 (6). bn+1 =   (6)
Therefore n must be 500000. Since n=0 is the first term, n=500000 is the 5000001 term. But does not affect the accuracy so there must be exactly 500000 terms in order to reach 6-digit accuracy.
Using this series, it takes an enormous amount of terms to simply get 6-digit accuracy. Therefore we should try to find another series that converges to /4 much faster. One possible way is through the formula below (7). 4tan-1(1/5) – tan-1(1/239) = /4 (7) To prove (7), we could use this formula (8) but the coefficient of 4 in front of the inverse tangent function causes a problem. tan-1(x) - tan-1(y) = tan-1  (8)
Therefore we must incorporate the double angle formula for tangent (9). tan 2x =  (9)
Substituting u = tan x (x = tan-1 u) into this formula we get the following: x =  or tan-1 u = (10)
2 tan-1 u =  ) (11)
This gives us a coefficient of only 2 however so we need to multiply both sides by 2. 4 tan-1 u = 2 ) (12)
Notice however that the right side of equation 12 is the same as the left side of equation 11 with a different value. Therefore we can substitute (2u/1-u2) for u in equation 11. Then we get an equation for changing an inverse tangent function with a coefficient of 1 into an inverse tangent function with a coefficient of 4.
Again to determine accuracy to a certain number of digits, in this case 6, we must utilize equation 5. Like in equation 6, bn+1 must be 1/1000001 to achieve 6-digit accuracy. But unlike equation 4, using this new series expansion we reach 6-digit accuracy much faster. It takes only 4 terms before we have 6-digit accuracy. 4   -  1/1000001 (18)
Since this series expansion for tan-1x allows us to reach higher accuracy much faster, it is the more efficient of the two series. It is therefore a better series expansion for calculating . Example: Calculation of Pi to 707-digit accuracy (like William Shanks): To determine this series to 707-digit accuracy we basically follow the same procedure as above. We find that we need only 504 terms before we can achieve 707-digit accuracy. 4  -  1x10-707 (19)
The History of the Calculation of Pi Anyone who knows the simple formula for the area of a circle (Area = pi times the square of the radius) can compute pi from that relationship (Alfeld 1). However, calculating it accurately using this method requires much time and patience. Still, this approach to computing was used for over 3500 years until the late 17th century when the much more efficient calculus-based series expansions became available. About two thousand years B.C., the first calculations of pi were made, probably by the Egyptians and Babylonians. A document entitled the Rhind Papyrus, dated 1650 BC, shows that the Egyptians obtained the value of pi to be (4/3)^4; the Babylonians earlier approximated a value of 3 1/8 for pi. At about the same time, it is believed that the Indians were using the square root of 10. These early approximations of pi were not very accurate as they were correct to only 1 decimal place (A Brief History 1). T Following in Archimedes’ footsteps in 150 AD was Ptolemy of Alexandria. He estimated the value of pi to be 377/120. The Chinese were a part of this pi race also. In 500 AD Tsi Ch’ung-Chi gave the value of pi at 355/113. These estimations are correct by 3 and 6 decimal places respectively (A Brief History 1). 
Other contributors include:
(used Archimedes’s method and a  -sided polygon)
After Van Ceulen, we encounter the method that we used to solve the problem in the previous pages. Wrongly entitled the Leibniz series, it was actually created by James Gregory. 
Using the two formulas above (like we did today), John Machin (c. 1706) calculated 100 decimal digits of Another interesting mathematical formula for pi was discovered during the European Renaissance by Wallis (1616-1703).
Here the two products continue indefinitely and they finally converge to the value of pi over 2 (A Brief History 1). William Shanks (c. 1807) determined the first 707 digits of After the invention of desktop calculators, D. F. Ferguson (c. 1947) raised the total to 808 (accurate) decimal digits. In fact, it was Ferguson who discovered the errors in Shanks' calculations (Carothers 1). The current record for most pi digits is held by Yasumasa Kanada and Daisuke Takahashi from the University of Tokyo with 51 billion digits of pi (51,539,600,000 decimal digits to be precise) (Alfeld 1). Now that’s a lot of pi. (The number on the cover is pi to 2,462 decimal places) Works Cited Alfeld, Peter. “Archimedes and the Computation of Pi.” 01 April 1999. University of Utah 05 December 2000 “A Brief History of Pi.” The Pi Project. Univ. of Texas. 05 December 2000
“Calculating Pi.” The Pi Project. Univ. of Texas. 05 December 2000 Carothers, Neal. “The Precomputer History of .” 05 December 2000 Download 27.44 Kb. Share with your friends:
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= 
…… (1)
+ 
+
……. (2)
= 
…….. (4)
) (13)
= tan-1 (120/119) (14)
= tan-1(1) = /4 (16)
- 
= /4 (17)
he next major step towards a more accurate value of pi was taken by the Greek mathematician Archimedes who decided to take up the problem in about 250 BC. He observed that area of a unit circle (a circle with radius 1) equals the exact value of pi. By first finding the area of a square inscribed in the circle, then a pentagon, and so on up through 96-sided polygon, Archimedes continuously found better and better approximations for pi. With each polygon having more sides (close to becoming a circle), he came closer and closer to finding an exact value of pi. He finally discovered that 3 10/71 < pi < 3 10/70. Many people often use the latter value, 22/7, when dealing with pi (when it does not require precise calculations). In fact, it is used so often that some people actually think that it is the exact value of pi (A Brief History 1).
. (Carothers 1)