Newton ’s

Newton’s
Approximation of Pi
By: Sarah Riffe
and Jen Watt
 Outline
• Who was Isaac Newton? What was
his life like?

• What is the history of Pi?

• What was Newton’s approximation of

Pi?
History of Isaac Newton
• 17th Century

– Shift of progress in math

– “relative freedom” of thought in

Northern Europe
The Life of Newton
• Born: Christmas
day 1642

• Died: 1727

• Raised by
grandmother
 Newton’s Education
• 1661
• Began at Trinity College of
Cambridge University

• 1660
• Charles II became King of England
• Suspicion and hostility towards
Cambridge
 Newton, the young man
• “single minded”
– Would not eat or sleep over an intriguing
problem

• Puritan
– Book of sins
 Newton’s Studies
• 1664
– Promoted to scholar at Trinity

• 1665-1666
– Plague
– Newton’s most productive years
 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
 Newton’s
Accomplishments
• 1668
– Finished master’s degree
– Elected fellow of Trinity College

• 1669
– Appointed Lucasian chair of
mathematics
 Newton’s
Accomplishments
• @ 1704
– Elected President of
the Royal Society

• 1705
– Knighted by Queen
Anne

• 1727
– Buried in Westminster
Abbey
 The History of Pi
• Archimedes’
classical method
– Using Polygons
with inscribed
And
Circumscribed
circles
– Found Pi between 223/71 and 22/7
• =3.14
 Important Dates of Pi
• 150 AD
– First notable value for Pi by Caludius
Ptolemy of Alexandria
– Pi = 3 8’30”
= 377/120
= 3.1416
• 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
• 1150 AD
– Bhaskara
– Pi = 3,927/1250
Pi = 22/7
Pi = 754/240
= 3.1416
• 1429 AD
– Al- Kashi
– Astronomer approximated Pi to 16
decimal places
• 1579 AD
– Francois Viete from France
– Approximated Pi to 9 decimal places
• 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 with
2^30th sides
• 1610 AD
– Ludolph Van Ceulen of the Netherlands
– Pi ~ 30 decimal places
– Used polygons with sides
62
2
• 1621 AD
– Willebrord Snell (Dutch)
– Able to get Ceulen’s 35th decimal place by only
2 side polygon
30
• 1630 AD
– Grienberger
– Pi to 39 decimal places
• 1671
– James Gregory from Scotland obtained
infinite series

x3 x5 x7
arctan x = x − + − + ...(− 1 ≤ x ≤ 1)
3 5 7
• 1699 AD
– Abraham Sharp
– Pi ~ 71 decimal places
• 1706 AD
– John Machin
– Pi ~ 100th decimal place
• 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
• 1767 AD
– Johan Heinrich Lambert
– Showed Pi is irrational
• 1794 AD
– Adrien-Marie Legendre
– Showed Pi-squared is irrational
• 1841 AD
– William Rutherford
– Calculated Pi to 208 places
• 1844 AD
– Zacharis Dase found Pi correct to 200
places using Gregory Series

π 1 1 1

= arctan   + arctan   + arctan  
2 2 5 8
• 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
• 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 Pi
Used Machin’s formula

π 1  1   1 
= 3 arctan   + arctan   + arctan  
4 4  20   1985 
• 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
• 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 Pi
to 250,000 decimal places on a STRETCH
computer
• 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 to
1 millionth place on CDC 7600

• 1981 AD
– Kazunori Miyoshi and Kazuhika Nakayma of
the University of Tsukuba
– Pi to 2 million and 38 decimal places in 137.30 hours on a
FACOM M-200 computer
• 1986 AD
– DH Bailey of NASA Ames Research Center
ran 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
 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
Information Already
known
2
 1
2

 x −  + ( y − 0) =
2 1
 2 2

or

x − x+
2 1 +y =1
2
4 4
 Solve for “y”

y = x1/ 2 (1 − x)1/ 2
1 1 2 1 3 5 4 7 5
= x (1 − x − x − x −
1/ 2
x − x − ...)
2 8 16 128 256
1 3/ 2 1 5/ 2 1 7/ 2 5 9/ 2 7 11/12
=x − x − x − x −
1/ 2
x − x − ...
2 8 16 128 256
Area (ABD) by fluxion

2 3/ 2 1  2 5 / 2  1  2 7 / 2  1  2 9 / 2 
x −  x  −  x  −  x  − ...
3 25  87  16  9 
2 3 / 2 1 5/ 2 1 7 / 2 1 9 / 2 5 11/ 2
= x − x − x − x − x − ...
3 5 28 72 704
 3 5
1
3/ 2
 1 5/ 2
 1
 = __, __  
1 1 1
  =   =   =
 __ ...
4  4  8 4  4 32

1 1 1 5 429
− − − .... − = .07677310678
12 160 3584 1441792 163208757248
 Area (ABD) by geometry
2 2
1  1 3 3
BD =   −   = =
2 4 16 4

1  1  3 
1
( )( )
Area( ∆DBC ) = BC x BD =  
2
=
3
2  4  4  32
 1
Area(sec tor ) = Area( semicircle)
3
11 2
=  ⋅π ⋅ r 
32 
1 1 1 
 2

=  π  
3  2  2  
π
=
24
 Area ( ABD) = Area(sec tor ) − Area ( ∆DBC )
π 3
= −
24 32

 3
π ≈ 24 .07677310678 +  = 3.141592668...

 32 