Newton's Approximation of Pi
Transcript of Newton's Approximation of Pi
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Newton’s Approximation of Pi
NewtonNewton’’s s Approximation of PiApproximation of Pi
By: Sarah By: Sarah RiffeRiffeand Jen Wattand Jen Watt
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Outline• Who was Isaac Newton? What was
his life like?
• What is the history of Pi?
• What was Newton’s approximation of Pi?
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History of Isaac Newton• 17th Century
– Shift of progress in math
– “relative freedom” of thought in Northern Europe
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The Life of Newton• Born: Christmas
day 1642
• Died: 1727
• Raised by grandmother
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Newton’s Education• 1661• Began at Trinity College of
Cambridge University
• 1660• Charles II became King of England• Suspicion and hostility towards
Cambridge
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Newton, the young man• “single minded”
– Would not eat or sleep over an intriguing problem
• Puritan– Book of sins
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Newton’s Studies• 1664
– Promoted to scholar at Trinity
• 1665-1666– Plague– Newton’s most productive years
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Newton’s Discoveries• 1665
– Newton’s “generalized binomial theorem”– led to method of fluxions
• 1666– Inverse method of fluxions– Began observations of rotation of
planets
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Newton’s Accomplishments
• 1668– Finished master’s degree– Elected fellow of Trinity College
• 1669– Appointed Lucasian chair of
mathematics
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Newton’s Accomplishments
• @ 1704– Elected President of
the Royal Society
• 1705– Knighted by Queen
Anne
• 1727– Buried in Westminster
Abbey
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The History of Pi• Archimedes’
classical method– Using Polygons with inscribedAndCircumscribedcircles
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– Found Pi between 223/71 and 22/7• =3.14
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Important Dates of Pi• 150 AD
– First notable value for Pi by CaludiusPtolemy of Alexandria
– Pi = 3 8’30”= 377/120= 3.1416
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• 480 AD– TSU Ch’ung-chih from China gave rational
approximation– Pi = 355/113
= 3.1415929• 530 AD
– Hindu mathematician Aryabhata– Pi = 62,832/20,000
= 3.1416
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• 1150 AD– Bhaskara– Pi = 3,927/1250Pi = 22/7Pi = 754/240
= 3.1416
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• 1429 AD– Al- Kashi– Astronomer approximated Pi to 16
decimal places • 1579 AD
– Francois Viete from France– Approximated Pi to 9 decimal places
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• 1585 AD– Adriaen Anthoniszoon– Rediscovered Chinese ratio 355/113– 377/120> Pi > 333/106
• 1593 AD– Adriaen Von Roomen– Found Pi to the 15th decimal place by
classical method using polygons with2^30th sides
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• 1610 AD– Ludolph Van Ceulen of the Netherlands– Pi ~ 30 decimal places– Used polygons with sides
• 1621 AD– Willebrord Snell (Dutch)– Able to get Ceulen’s 35th decimal place by only
side polygon
622
302
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• 1630 AD– Grienberger– Pi to 39 decimal places
• 1671– James Gregory from Scotland obtained
infinite series
( )11...753
arctan753
≤≤−+−+−= xxxx
xx
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• 1699 AD– Abraham Sharp– Pi ~ 71 decimal places
• 1706 AD– John Machin– Pi ~ 100th decimal place
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• 1719 AD– De Lagny of France– Pi ~ 112 decimal places
• 1737 AD– William Jones from England– First to use Pi symbol for ratio of the
circumference to the diameter
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• 1767 AD– Johan Heinrich Lambert– Showed Pi is irrational
• 1794 AD– Adrien-Marie Legendre– Showed Pi-squared is irrational
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• 1841 AD– William Rutherford– Calculated Pi to 208 places
• 1844 AD– Zacharis Dase found Pi correct to 200
places using Gregory Series
+
+
=
81
arctan51
arctan21
arctan2π
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• 1853 AD– Rutherford returns– Finds Pi to 400 decimal places
• 1873 AD– William Shanks from England– Pi to 707 decimal places
• 1882 AD– F. Lindeman– Shows Pi is transcendental
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• 1948– D.F. Ferguson of England
• Finds errors with Shanks value of Pi starting with the 528th decimal place
• Gives correct value to the 710th place– J.W. Wrench Jr.
• Works with Ferguson to find 808th place for PiUsed Machin’s formula
+
+
=
19851
arctan201
arctan41
arctan34π
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• 1949 AD– Electronic computer – The ENIAC
• Compute Pi to the 2,037th decimal places
• 1959 AD– Fancois Genuys from Paris– Compute Pi to 16,167 decimal places with IBM 704
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• 1961 AD– Wrench and Shanks of Washington D.C.– compute Pi to 100,265th
using IBM 7090• 1966 AD
– M. Jean Guilloud and co-workers– attained approximation for Pito 250,000 decimal places on a STRETCH computer
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• 1967 AD– M. Jean Guilloud and coworkers– found Pi to the 500,000 places on a CDC 6600
• 1973– M. Jean Guilloud and coworkers found Pi to1 millionth place on CDC 7600
• 1981 AD– Kazunori Miyoshi and Kazuhika Nakayma ofthe University of Tsukuba– Pi to 2 million and 38 decimal places in 137.30 hours on a
FACOM M-200 computer
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• 1986 AD– DH Bailey of NASA Ames Research Centerran a Cray-2 supercomputer for 28 hours
• Got Pi to 29,360,000 decimal places
– Yasamasa Kanada from University of Tokyo• Used NEC SX-2 super computer to compute Pi
to 134,217,700 decimal places
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Purpose to Continue to Compute Pi
• See if digits of Pi start to repeat– Possible normalcy of Pi
• Valuable in computer science for designing programs
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Information Already known
( )2
22
21
021 =−+
− yx
or
41
41 22 =++− yxx
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Solve for “y”
...2567
1285
161
81
21
...)2567
1285
161
81
21
1(
)1(
12/112/92/72/52/32/1
54322/1
2/12/1
−−−−−−=
−−−−−−=
−=
xxxxxx
xxxxxx
xxy
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Area (ABD) by fluxion
...7045
721
281
51
32
...92
161
72
81
52
21
32
2/112/92/72/52/3
2/92/72/52/3
−−−−−=
−
−
−
−
xxxxx
xxxx
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...__321
41
41
____,81
41
41
52/532/3
=
=
=
=
80767731067.481632087572
429....
14417925
35841
1601
121
=−−−−
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Area (ABD) by geometry
43
163
41
21
22
==
−
=BD
( ) ( )32
343
41
21
21
)( =
==∆ BDxBCDBCArea
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24
21
21
31
21
31
)(31
)(sec
2
2
π
π
π
=
=
⋅⋅=
=
r
semicircleAreatorArea
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323
24
)()(sec)(
−=
∆−=
π
DBCAreatorAreaABDArea
...141592668.332
380767731067.24 =
+≈π