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

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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

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World History 1609: Galileo launched modern day astronomy: Planets revolve around the sun not the Earth1609: 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.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 Japan1637: 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.1642: Puritans under Oliver Cromwell won campaign against monarchy and Cromwell assumed control of English government.

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World History 1649: King Charles I was beheaded by Cromwell’s government1649: King Charles I was beheaded by Cromwell’s government 1658: Cromwell died1658: Cromwell died 1660: Charles II placed on thrown: The beginning of the Restoration in Britain1660: Charles II placed on thrown: The beginning of the Restoration in Britain

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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

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Mathematical History Early 1600s: John Napier and Henry Briggs introduced, perfected and exploited logarithms.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,1637: Rene Descartes wrote Discours de la methode: a landmark in the history of philosophy. Appendix: La Geometrie first published account of analytical geometry,

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Mathematical History Blaise Pascal : 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

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Mathematical History : Pierre de Fermat created analytical geometry different from Descartes. Laid foundation for probability theory : Pierre de Fermat created analytical geometry different from Descartes. Laid foundation for probability theory Fermat’s last theorem: a n +b n =c n no known whole number solution for n>3.Fermat’s last theorem: a n +b n =c n no known whole number solution for n>3.

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Isaac 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 age of 3Mother left him to live with grandmother at age of 3 Had respectable grammar school education consisting mostly of Latin and Greek.Had respectable grammar school education consisting mostly of Latin and Greek. Kept mostly to himself, reading and building many miniature devicesKept mostly to himself, reading and building many miniature devices

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Newton’s Inventions +

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Sundials Lanterns attached to kites

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Isaac Newton 1661: Newton went to Trinity College, Cambridge1661: Newton went to Trinity College, Cambridge Met Cambridge Professor Isaac Barrow who directed Newton to the major sources of contemporary mathematics.Met Cambridge Professor Isaac Barrow who directed Newton to the major sources of contemporary mathematics. 1664: Promoted to Scholar at Cambridge1664: Promoted to Scholar at Cambridge Newton’s “wonderful years” when most his work was completed was during the two plague years.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 : Newton wrote De Analysi regarding fluxonal ideas; precursor to calculus. Wasn’t published until 1711

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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.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.Took over for Barrow as Lucasian professor lecturing on mathematics with minimal attendance. Performed numerous experiments on himself to study optics such as: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

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Newton’s Binomial Theorem First great mathematical discoveryFirst great mathematical discovery Theorem stated that given an binomial P + PQ raised to the power m/n we have:Theorem stated that given an binomial P + PQ raised to the power m/n we have:

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Newton’s B. Example From the generalized equation above, we get:

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Rules from De Analysi If The the area under the curve is Area ABD Where x=AB and y=BD

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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“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 isExample: The area under is

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Newton’s Approximation of π

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Area (ABD) by FluxionsArea (ABD) by Fluxions Evaluated at, we get the following from the first nine terms:Evaluated at, we get the following from the first nine terms:

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Newton’s Approximation of π Area (ABD) by GeometryArea (ABD) by Geometry By Pythagorean Theorem, givenBy Pythagorean Theorem, given Δ DBC, with length BC=1/4 and length CD, the radius = ½, we have Hence,

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Newton’s Approximation of π Area (sector ACD) = Area (semicircle)Area (sector ACD) = Area (semicircle) Due to the fact that

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Newton’s Approximation of π Equating this to the result found byEquating this to the result found by Newton’s fluxion method and Rearranging for π, we get:

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Newton’s Approximation of π Q.E.D.

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Video Rap

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Questions to Ponder How do you think Newton was able to calculate such precise approximations without the use of a calculator?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?Do you think Newton’s unusual upbringing had anything to do with his future contributions to math and physics?

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