Measuring%20the%20Circle:%20The%20Story%20of

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Measuring the CircleThe Story of

• Eric Wishnie
• Seth Coldsmith

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Formal Definition

• The ratio between a circles circumference and
  diameter.
• The ratio between a circles area and the square
  of its radius.
• first used by William Jones, 1706
• Adopted by Leonhard Euler in publications 1730s,
  40s

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Discovery as a Constant

• First considered only as 3.
• Egyptians, circa 1650 B.C., recognized a constant
  when computing the area of a circle

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Archimedes circa 240 B.C.

• Used polygons and circles to estimate .
• Involved the use of two similar methods.
• Inscribed polygons
• Circumscribed polygons
• Provided an upper limit of 3.142857143 and a
  lower limit of 3.14084507 for .
• Example

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Other advances after Archimedes

• Claudius Ptolemy circa 150 A.D. used his table of
  chords to estimate to be
• Chinese scholar Zu Chongzi circa 480 A.D. used
• for it, but his methods are unknown. He later
  worked out to be between 3.141596 and
  3.141597.

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Aryabhata

• First to use expression to calculate
• a length of one side of an inscribed polygon
  with n sides
• b length of one side of an inscribed polygon
  with 2n sides

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Brahmagupta circa 650 A.D.

• Also used method of inscribing polygons with
  doubling numbers of sides to estimate the value
  of .
• Found the values of
• From these results he concluded that was
  converging to

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Different Sequences Used

• John Wallis in 1650 A.D. proved that /2
  2/1 x 2/3 x 4/3 x 4/5 x 6/5 x 6/7 x
• Viscount Brouncker proposed a few years earlier
  the sequence
• Neither sequences were really used.

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Timeline of , Part I

• c. 1650 BCE Egyptians estimate pi as
• 240 BCE Archimedes uses polygons and circles to
  estimate upper and lower bounds of , method
  widely used later in Europe
• c. 530 CE Aryabhata develops first algorithmic
  method to calculate
• c. 650 CE Brahmagupta deduces as converging
  to
• c. 1650 CE First numerical methods of
  calculating developed as sequences
• 1765 CE Johann Heinrich Lambert proves is
  irrational
• 1949 ENIAC (Electronic Numerical Integrator and
  Computer), computes to 2035 decimal places,
  70 hours

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Timeline of , Part II

• 1987 University of Tokyo, under Prof. Yasumasa
  Kanada, calculates to 134,217,000 digits on
  NEC supercomputer
• 1991 Gregory and David Chudnovsky calculate
  to 2,260,321,336 decimal places on home-built
  supercomputer, 250 hours
• 1999 Prof. Kanada calculates again, achieves
  to 206,158,430,000 decimal places
• 2002 Takahashi Kanada calculates to 1.2411
  trillion digits

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References

• The Timeline of Pi. http//people.bath.ac.uk/lr2
  26/timeline.html
• Berlingholl and Gouvea
• Katz, Victor J. A History of Mathematics Brief
  Edition. Pearson/Wesley 2004
• Harris, Herman H. Jr., The History and
  Calculation of Pi. Emporia State Research
  Studies, Volume 8, Number 1 Graduate Division of
  the Kansas State Teachers College, Emopria
  Kansas, September 1959