Newton’s Approximation of pi Kimberly Cox, Matt Sarty, Andrew Wood.

Newton’s Approximation of Newton’s Approximation of pi pi

Kimberly Cox, Matt Sarty, Andrew WoodKimberly Cox, Matt Sarty, Andrew Wood

World HistoryWorld History

1601: William Shakespeare published his play 1601: William Shakespeare published his play Hamlet, Prince of DenmarkHamlet, Prince of Denmark

1605: Cervantes wrote monumental Don 1605: Cervantes wrote monumental Don Quixote the most influential piece of lit. to Quixote the most influential piece of lit. to come from the Spanish Golden Age.come from the Spanish Golden Age.

1607: Jamestown, Va. Settled by British. 1607: Jamestown, Va. Settled by British. Started the European Colonization of N. Started the European Colonization of N. AmericaAmerica

1608: Quebec City, known as New France was 1608: Quebec City, known as New France was settled by Samuel de. Champlainsettled by Samuel de. Champlain


• 1609: Galileo launched modern day astronomy: 1609: Galileo launched modern day astronomy: Planets revolve around the sun not the EarthPlanets revolve around the sun not the Earth

• 1633: Galileo faced the inquisition for ideas of 1633: Galileo faced the inquisition for ideas of astronomy and was named a heretic by the church astronomy and was named a heretic by the church in Rome.in Rome.

• 1637: Massacre of thousands of Japanese 1637: Massacre of thousands of Japanese Christians, beginning of period of National Isolation Christians, beginning of period of National Isolation in Japanin Japan

• 1642: Puritans under Oliver Cromwell won 1642: Puritans under Oliver Cromwell won campaign against monarchy and Cromwell assumed campaign against monarchy and Cromwell assumed control of English government. control of English government.


• 1649: King Charles I was beheaded by 1649: King Charles I was beheaded by Cromwell’s governmentCromwell’s government

• 1658: Cromwell died1658: Cromwell died

• 1660: Charles II placed on thrown: The 1660: Charles II placed on thrown: The beginning of the Restoration in Britain beginning of the Restoration in Britain

Mathematical HistoryMathematical HistoryFrancois Viete:Francois Viete:

In 1590 published In In 1590 published In Artem analyticam Artem analyticam isogage- The Analytic isogage- The Analytic Art which mentioned an Art which mentioned an approximation of pi and approximation of pi and used letters to represent used letters to represent quantities in an quantities in an equationequation

Ex: D in R- D in E Ex: D in R- D in E aequabitur A quad aequabitur A quad means DR-DE=Ameans DR-DE=A22

Mathematical HistoryMathematical History

• Early 1600s: John Napier and Henry Briggs Early 1600s: John Napier and Henry Briggs introduced, perfected and exploited introduced, perfected and exploited logarithms.logarithms.

• 1637: Rene Descartes wrote Discours de la 1637: Rene Descartes wrote Discours de la methode: a landmark in the history of methode: a landmark in the history of philosophy. Appendix: La Geometrie first philosophy. Appendix: La Geometrie first published account of analytical geometry,published account of analytical geometry,


Blaise PascalBlaise Pascal

1623-1662: Started contributing 1623-1662: Started contributing to math at age 14. to math at age 14.

Invented calculating machine: Invented calculating machine: precursor to modern computersprecursor to modern computers

Famous for Pascal’s triangle used Famous for Pascal’s triangle used in Binomial theoremin Binomial theorem

Later switched studies to theologyLater switched studies to theology


• 1601-1665: Pierre de Fermat created 1601-1665: Pierre de Fermat created analytical geometry different from Descartes. analytical geometry different from Descartes. Laid foundation for probability theoryLaid foundation for probability theory

• Fermat’s last theorem: aFermat’s last theorem: ann +b +bnn=c=cnn no known no known whole number solution for n>3.whole number solution for n>3.

Isaac NewtonIsaac Newton

• Born Christmas day 1642Born Christmas day 1642

• Father died shortly before his birthFather died shortly before his birth

• Mother left him to live with grandmother at Mother left him to live with grandmother at age of 3age of 3

• Had respectable grammar school education Had respectable grammar school education consisting mostly of Latin and Greek.consisting mostly of Latin and Greek.

• Kept mostly to himself, reading and building Kept mostly to himself, reading and building many miniature devicesmany miniature devices

Newton’s InventionsNewton’s Inventions

+

Newton’s InventionsNewton’s Inventions

Sundials Lanterns attached to kites


• 1661: Newton went to Trinity College, Cambridge1661: Newton went to Trinity College, Cambridge

• Met Cambridge Professor Isaac Barrow who Met Cambridge Professor Isaac Barrow who directed Newton to the major sources of directed Newton to the major sources of contemporary mathematics.contemporary mathematics.

• 1664: Promoted to Scholar at Cambridge1664: Promoted to Scholar at Cambridge

• Newton’s “wonderful years” when most his work Newton’s “wonderful years” when most his work was completed was during the two plague years.was completed was during the two plague years.

• 1669: Newton wrote De Analysi regarding fluxonal 1669: Newton wrote De Analysi regarding fluxonal ideas; precursor to calculus. Wasn’t published ideas; precursor to calculus. Wasn’t published until 1711 until 1711


• 1668: Newton elected a fellow at 1668: Newton elected a fellow at Trinity College allowing him to stay Trinity College allowing him to stay at the college with financial support at the college with financial support as long as he took holy vows and as long as he took holy vows and remained unmarried.remained unmarried.

• Took over for Barrow as Lucasian Took over for Barrow as Lucasian professor lecturing on mathematics professor lecturing on mathematics with minimal attendance.with minimal attendance.

• Performed numerous experiments Performed numerous experiments on himself to study optics such as:on himself to study optics such as:

- staring at the sun for extended - staring at the sun for extended periods of time and examining the periods of time and examining the spots in his eyesspots in his eyes

- pressing eye with small stick to - pressing eye with small stick to study the effect this had on his study the effect this had on his visionvision

Newton’s Binomial Newton’s Binomial TheoremTheorem

• First great mathematical discoveryFirst great mathematical discovery

• Theorem stated that given an binomial P + Theorem stated that given an binomial P + PQ raised to the power m/n we have: PQ raised to the power m/n we have:

€

(P+PQ)m / n = Pm / n +m

nAQ+

m − n

2nBQ+

m − 2n

3nCQ+

m − 3n

4nDQ+...

€

A = Pm / n

€

B =m

nAQ =

m

nPm / nQ

€

C =

m

n

⎛

⎝ ⎜

⎞

⎠ ⎟m

n−1

⎛

⎝ ⎜

⎞

⎠ ⎟

2Pm / nQ2

€

D =

m

n

⎛

⎝ ⎜

⎞

⎠ ⎟m

n−1

⎛

⎝ ⎜

⎞

⎠ ⎟m

n− 2

⎛

⎝ ⎜

⎞

⎠ ⎟

3× 2Pm / nQ3

Newton’s B. ExampleNewton’s B. Example

€

1− x =1−1

2x −

1

8x 2 −

1

16x 3 −

5

128x 4 −

7

256x 5 − ...

€

1+Q( )m / n

=1+m

nQ+

m

n

⎛

⎝ ⎜

⎞

⎠ ⎟m

n−1

⎛

⎝ ⎜

⎞

⎠ ⎟

2Q2 +

m

n

⎛

⎝ ⎜

⎞

⎠ ⎟m

n−1

⎛

⎝ ⎜

⎞

⎠ ⎟m

n− 2

⎛

⎝ ⎜

⎞

⎠ ⎟

3× 2Q3 + ...

From the generalized equation above, we get:

Rules from De AnalysiRules from De Analysi

€

axm

n = yIf

The the area under the curve is

€

an

m + nxm+n

n = Area ABD

Where x=AB and y=BD

Rules from De AnalysiRules from De Analysi

• ““If the Value of y be made up of several If the Value of y be made up of several Terms, the Area likewise shall be made up of Terms, the Area likewise shall be made up of the Areas which result from every one of the the Areas which result from every one of the terms.” – Rule 2terms.” – Rule 2

• Example: The area under Example: The area under isis

€

y = x 2 + x 3 / 2

€

1

3x 3 +

2

5x 5 / 2

Newton’s Newton’s Approximation of πApproximation of π

€

y = x − x 2 = x1

2 (1− x)1

2

= x1/ 2 −1

2x 3 / 2 −

1

8x 5 / 2 −

1

16x 7 / 2 −

5

128x 9 / 2 −

7

256x11/ 2 − ...


• Area (ABD) by FluxionsArea (ABD) by Fluxions

• Evaluated at , we get the following Evaluated at , we get the following from the first nine terms:from the first nine terms:

€

2

3x 3 / 2 −

1

2

2

5x 5 / 2 ⎛

⎝ ⎜

⎞

⎠ ⎟−

1

8

2

7x 7 / 2 ⎛

⎝ ⎜

⎞

⎠ ⎟−

1

16

2

9x 9 / 2 ⎛

⎝ ⎜

⎞

⎠ ⎟− ...

=2

3x 3 / 2 −

1

5x 5 / 2 −

1

28x 7 / 2 −

1

72x 9 / 2 −

5

704x11/ 2 − ...

€

x =1

4

€

1

12−

1

160−

1

3584−

1

36864−

1

1441792− ...−

429

163208757248= 0.07677310678


• Area (ABD) by GeometryArea (ABD) by Geometry

• By Pythagorean Theorem, given By Pythagorean Theorem, given

ΔDBC, with length BC=1/4 and length ΔDBC, with length BC=1/4 and length

CD, the radius = ½, we haveCD, the radius = ½, we have

Hence, Hence,

€

Area DBC( ) =1

2BC_____ ⎛ ⎝ ⎜

⎞ ⎠ ⎟× BD

______ ⎛ ⎝ ⎜

⎞ ⎠ ⎟=

1

2

1

4

⎛

⎝ ⎜

⎞

⎠ ⎟

3

4

⎛

⎝ ⎜

⎞

⎠ ⎟€

BD =1

2

⎛

⎝ ⎜

⎞

⎠ ⎟2

−1

4

⎛

⎝ ⎜

⎞

⎠ ⎟2 ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

1

2

=3

16=

3

4


• Area (sector ACD) = Area (semicircle)Area (sector ACD) = Area (semicircle)

• Due to the fact that <BCD=60°, or 1/3Due to the fact that <BCD=60°, or 1/3

of the 180° forming the semicircle.of the 180° forming the semicircle.

• Area (ABD) = Area (sector ACD) – Area (ΔDBC)Area (ABD) = Area (sector ACD) – Area (ΔDBC)

= =

€

1

3

€

=1

3

1

2πr2 ⎛

⎝ ⎜

⎞

⎠ ⎟=π

24

€

π24

−3

32


• Equating this to the result found byEquating this to the result found by

Newton’s fluxion method andNewton’s fluxion method and

Rearranging for π, we get:Rearranging for π, we get:

€

π ≈24 0.07677310678 +3

32

⎛

⎝ ⎜

⎞

⎠ ⎟= 3.141592668


Q.E.D.Q.E.D.

Video RapVideo Rap

• http://www.youtube.com/watch?http://www.youtube.com/watch?v=BjypFm58Ny0v=BjypFm58Ny0

Questions to PonderQuestions to Ponder

• How do you think Newton was able to How do you think Newton was able to calculate such precise approximations without calculate such precise approximations without the use of a calculator?the use of a calculator?

• Do you think Newton’s unusual upbringing Do you think Newton’s unusual upbringing had anything to do with his future had anything to do with his future contributions to math and physics?contributions to math and physics?

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Newton’s Approximation of pi Kimberly Cox, Matt Sarty, Andrew Wood.

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