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Newton’s Approximation of pi

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.

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Newton’s Approximation of pi

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

  2. 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

  3. 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.

  4. 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

  5. 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

  6. 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,

  7. 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

  8. 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.

  9. 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

  10. Newton’s Inventions +

  11. Newton’s Inventions Lanterns attached to kites Sundials

  12. 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

  13. 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

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

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

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

  17. 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 is

  18. Newton’s Approximation of π

  19. Newton’s Approximation of π • Area (ABD) by Fluxions • Evaluated at , we get the following from the first nine terms:

  20. 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 Hence,

  21. Newton’s Approximation of π • Area (sector ACD) = Area (semicircle) • Due to the fact that <BCD=60°, or 1/3 of the 180° forming the semicircle. • Area (ABD) = Area (sector ACD) – Area (ΔDBC) =

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

  23. Newton’s Approximation of π Q.E.D.

  24. Video Rap • http://www.youtube.com/watch?v=BjypFm58Ny0

  25. 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?

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