Measuring the Circle: The Story of Eric Wishnie Seth Coldsmith

Formal Definition The ratio between a circle’s circumference and diameter. The ratio between a circle’s area and the square of its radius. “ ” first used by William Jones, 1706 –Adopted by Leonhard Euler in publications 1730s, ’40s

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

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 and a lower limit of for. Example:

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 and

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

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

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.

Timeline of, Part I c 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 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

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 trillion digits

References “The Timeline of Pi.” 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