Transcript Newton`s Approximation of pi - Mathematics & Computer Science

Newton’s
Approximation of pi
Kimberly Cox, Matt Sarty, Andrew Wood
World History
 1601: William Shakespeare published his play
Hamlet, Prince of Denmark
 1605: Cervantes wrote monumental Don Quixote the
most influential piece of lit. to come from the Spanish
Golden Age.
 1607: Jamestown, Va. Settled by British. Started the
European Colonization of N. America
 1608: Quebec City, known as New France was settled
by Samuel de. Champlain
World History
• 1609: Galileo launched modern day astronomy: Planets
revolve around the sun not the Earth
• 1633: Galileo faced the inquisition for ideas of astronomy
and was named a heretic by the church in Rome.
• 1637: Massacre of thousands of Japanese Christians,
beginning of period of National Isolation in Japan
• 1642: Puritans under Oliver Cromwell won campaign
against monarchy and Cromwell assumed control of
English government.
World History
• 1649: King Charles I was beheaded by Cromwell’s
government
• 1658: Cromwell died
• 1660: Charles II placed on thrown: The beginning of
the Restoration in Britain
Mathematical
History
Francois Viete:
 In 1590 published In Artem
analyticam isogage- The
Analytic Art which
mentioned an approximation
of pi and used letters to
represent quantities in an
equation
 Ex: D in R- D in E
aequabitur A quad means
DR-DE=A2
Mathematical
History
• Early 1600s: John Napier and Henry Briggs
introduced, perfected and exploited logarithms.
• 1637: Rene Descartes wrote Discours de la methode: a
landmark in the history of philosophy. Appendix: La
Geometrie first published account of analytical
geometry,
Mathematical
History
Blaise Pascal
1623-1662: Started contributing to
math at age 14.
Invented calculating machine:
precursor to modern computers
Famous for Pascal’s triangle used in
Binomial theorem
Later switched studies to theology
Mathematical
History
• 1601-1665: Pierre de Fermat created analytical
geometry different from Descartes. Laid foundation for
probability theory
• Fermat’s last theorem: an +bn=cn no known whole
number solution for n>3.
Isaac Newton
• Born Christmas day 1642
• Father died shortly before his birth
• Mother left him to live with grandmother at age of 3
• Had respectable grammar school education consisting
mostly of Latin and Greek.
• Kept mostly to himself, reading and building many
miniature devices
Newton’s Inventions
+
Newton’s Inventions
Sundials
Lanterns attached to kites
Isaac Newton
• 1661: Newton went to Trinity College, Cambridge
• Met Cambridge Professor Isaac Barrow who directed
Newton to the major sources of contemporary
mathematics.
• 1664: Promoted to Scholar at Cambridge
• Newton’s “wonderful years” when most his work was
completed was during the two plague years.
• 1669: Newton wrote De Analysi regarding fluxonal ideas;
precursor to calculus. Wasn’t published until 1711
Isaac Newton
• 1668: Newton elected a fellow at Trinity
College allowing him to stay at the college
with financial support as long as he took
holy vows and remained unmarried.
• Took over for Barrow as Lucasian
professor lecturing on mathematics with
minimal attendance.
• Performed numerous experiments on
himself to study optics such as:
- staring at the sun for extended periods of
time and examining the spots in his eyes
- pressing eye with small stick to study the
effect this had on his vision
Newton’s Binomial
Theorem
• First great mathematical discovery
• Theorem stated that given an binomial P + PQ raised
to the power m/n we have:
(PPQ)
AP

m/n
P
m/n
m
mn
m 2n
m  3n
 AQ
BQ
CQ
DQ...
n
2n
3n
4n
m /n
m
m m/n
B  AQ P Q
n
n

m m 
  1
n n  m / n 2
C
P Q
2
m m m 
  1  2
n n n
 m / n 3
D
P Q
3 2
Newton’s B. Example
1 Q
m/n
m m 
m m m 
  1
  1  2
m
n n  2 n n n
 3
1 Q 
Q 
Q  ...
n
2
32
From the generalized equation above, we get:
1
1 2 1 3 5 4
7 5
1 x 1 x  x  x 
x 
x  ...
2
8
16
128
256
Rules from De Analysi
If

m
n
ax  y
Where x=AB
and y=BD
The the area under the
curve is
an
x
mn
m n
n
 Area ABD
Rules from De Analysi
• “If the Value of y be made up of several Terms, the
Area likewise shall be made up of the Areas which
result from every one of the terms.” – Rule 2
• Example: The area under
yx x
1 3 2 5/2
x  x
3
5


2
3/2
is
Newton’s
Approximation of π
1
2
y  x  x 2  x (1 x)
x
1/ 2
1
2
1 3/2 1 5/2 1 7/2
5 9/2
7 11/ 2
 x  x  x 
x 
x  ...
2
8
16
128
256

Newton’s
Approximation of π
• Area (ABD) by Fluxions
2 3 / 2 1 2 5 / 2  1 2 7 / 2  1 2 9 / 2 
x   x   x   x  ...
 8 7
 16 9

3
2 5
2
1
1
1 9/2
5 11/ 2
 x3/2  x5/2  x7/2 
x 
x  ...
3
5
28
72
704
1
• Evaluated at x  , we get the following from the first
4
nine terms:
1
1
1
1
1
429




 ...
12 160 3584 36864 1441792
163208757248
 0.07677310678

Newton’s
Approximation of π
• Area (ABD) by Geometry
• By Pythagorean Theorem, given
ΔDBC, with length BC=1/4 and length
CD, the radius = ½, we have
1
1 2 1 2 2
3
3
BD  








2  4  
16
4


Hence,

1 _____  ______  1 1  3 
AreaDBC  BC   BD    
2     2 4  4 
Newton’s
Approximation of π
1
• Area (sector ACD) = Area (semicircle)
  3
1 1
  r 2 
 24
3 2
  fact that <BCD=60°, or 1/3
• Due to the

of the 180° forming the semicircle.
• Area (ABD) = Area (sector ACD) – Area (ΔDBC)
=

24

3
32

Newton’s
Approximation of π
• Equating this to the result found by
Newton’s fluxion method and
Rearranging for π, we get:

3 
  240.07677310678  3.141592668
32 

Newton’s
Approximation of π
Q.E.D.
Video Rap
• http://www.youtube.com/watch?v=BjypFm58Ny0
Questions to Ponder
• How do you think Newton was able to calculate such
precise approximations without the use of a calculator?
• Do you think Newton’s unusual upbringing had
anything to do with his future contributions to math
and physics?