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=A 2

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: a n number solution for n>3.

+b n =c n no known whole

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:

A B

(

P



PQ

)

m

/

n

 

P m m n

/

n AQ

 

P m

/

n m n



P m m n

/

n AQ



m



n

2

n C BQ

 

m

 2

n

3

n



m n

 

m n

2

CQ



m

 4

n

 1  

P m

/

n Q

2 3

n Q



D

 

m n

  

m n

 1  

m n

3  2

DQ

  2  

P m

/

n Q

3 ...

 



Newton’s B. Example

 1 

Q



m

/

n

 1 

m Q



n



m n

  

m n

2  1  

Q

2  

m n

  

m n

 1  

m n

3  2  2  

Q

3  ...

From the generalized equation above, we get: 1 

x

 1  1 2

x

 1 8

x

2  1 16

x

3  5 128

x

4  7 256

x

5  ...



Rules from De Analysi

 If

ax n m



y

Where x=AB and y=BD The the area under the curve is

m an



n x 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

x

2 

x

3 / 2 1  3

x

3  2 5

x

5 / 2 

Newton’s Approximation of π



y



x



x

2  1

x

2 (1 

x

) 2 1 

x

1/ 2  1 2

x

3 / 2  1 8

x

5 / 2  1 16

x

7 / 2  5 128

x

9 / 2  7 256

x

11/ 2  ...

 

Newton’s Approximation of π

• Area (ABD) by Fluxions 2 3

x

3 / 2  1 2  2 5

x

5 / 2  1 8  2 7

x

7 / 2  1 16  2 9

x

9 / 2   ...

 2 3

x

3 / 2  1 5

x

5 / 2  1 28

x

7 / 2  1 72

x

9 / 2  5 704

x

11/ 2  ...

nine terms:

x

 1 12   1 160  1 3584  1 1 4 36864  1 1441792  ...

 429 163208757248

Newton’s Approximation of π

• Area (ABD) by Geometry • By Pythagorean Theorem, given Δ DBC, with length BC=1/4 and length  CD, the radius = ½, we have

BD

     1  2  2  Hence,   1 4  2    1 2     2 1  _____

BC

 3 16   ______

BD

 4 3 2 1  1 4     4 3    

Newton’s Approximation of π

• Area (sector ACD) = Area (semicircle) 3  1 3  1 2 

r

2   24   ° , or 1/3 of the 180 ° forming the semicircle.

• Area (ABD) = Area (sector ACD) – Area ( Δ DBC) =  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:    24 0.07677310678

 3 32   3.141592668

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?