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Three Ways to Compute Pi

This presentation delves into three pivotal methods for computing the mathematical constant pi (π): Archimedes' use of 96-sided polygons, Machin's formula involving arctangents, and the innovative approach by Bailey, Borwein, and Plouffe. It discusses the historical context of these calculations, the importance of rigorous proof, and the challenges faced in understanding these mathematical innovations. We explore the legacy of these mathematicians and how their contributions have impacted the computation of π throughout history.

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Three Ways to Compute Pi

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  1. Three Ways to Compute Pi • Archimedes • Machin • Bailey, Borwein, and Plouffe V. Frederick Rickey 10-09-10 ARL

  2. Archimedes uses regular polygons with 96 sides to compute 3 + 10/71 < π < 3 + 10/70.

  3. Kepler’s Intuition, 1615

  4. Prove Machin's identity:

  5. Fortunately, a hint:

  6. Do you understand the details? • Is the proof correct? • Not so fast. You know tan (π/4) =1, so all you proved is 1 = 1.

  7. This is correct, inelegant, uninspiring mathematics. • This is not the way to do mathematics. • You must explain where the 1/5 comes from. • Time for some historical research.

  8. D. E. Smith, Source Book (1929)

  9. Use Gregory’s Series for the arctangent and Machin’s formula to get:

  10. Off to the Library William Jones by William Hogarth, 1740

  11. Francis Maseres 1731 –1824

  12. Machin’s Formulaand Proof first published here. But 1/5 still not explained.

  13. Charles Hutton:A Treatise on Mensuration (1770) The Idea • Want a small arc whose tangent is easy to compute • Tan 45o = 1 • Easy to find tan 2x, given tan x. • Repeat till tan 2nx ≈ 1. Now Experiment • Try tan x = 1/5 • Then tan 4x = 120/119 ≈ 1.

  14. Now let’s compute Plan: Compute 1/5n for odd n. Compute positive terms by dividing by the appropriate odd number, then add. Compute negative terms: divide, then add. Subtract sum of negatives from sum of positives and multiply by 16. Repeat for the other series.

  15. For the second series, divide repeatedly by 2392 = 57121:

  16. Thus π≈ 3.141,592,653,589,238,426

  17. Was Machin an Idiot Savant? • Tutored Brook Taylor before Cambridge. • Taylor announced “Taylor’s Theorem” to Machin. • His work on π attracted Newton’s attention. • Elected FRS in 1710. • Edited the Commerciumepistolicumwith Halley, Jones, 1712. • Became Gresham Professor of Astronomy in 1713. • A proposition of Machin on nodes of the moon is included in the third edition of Newton’s Principia • John Conduitt wrote: “Sr. I. told me that Machin understood his Principia better than anyone.”

  18. Bailey, Borwein, Plouffe, 1997 • The ten billionth hexadecimal digit of π is 9. • Discovered by “inspired guessing and extensive searching.”

  19. Questions ?

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