Pi

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playing around with pi

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Pi

  1. 1. Comparative analysis of Algorithms to Calculate π Presented by Rohit Kumar
  2. 2. Trivia <ul><li>The Greek letter π, often spelled out pi in text, was adopted for the number from the Greek word for perimeter &quot;περίμετρος&quot;, probably by William Jones in 1706 . </li></ul><ul><li>According to the Bible, π = 3 ! </li></ul><ul><li>“ Pi Day” is celebrated on March 14 th (3-14) which is also the Birthday of Albert Einstein. </li></ul><ul><li>Current World Record : Chou Lu of China recited 67,890 decimal places of pi on 20 November 2005. </li></ul><ul><li>The value of pi has been calculated to more than a trillion (10 12 ) digits . </li></ul>
  3. 3. What is π ? <ul><li>Pi is the Ratio of any circle’s circumference to its diameter in Euclidean Geometry. </li></ul><ul><li>Numerical value of π : </li></ul><ul><li>3.14159 26535 89793 23846 26433 83279… </li></ul><ul><li>Pi is an irrational no and hence its decimal expansion never ends or repeats. </li></ul>
  4. 4. Calculating π Brief History <ul><li>Earliest known approximations(around 1900 BC) : </li></ul><ul><li>Babylonia - 25/8 </li></ul><ul><li>Egypt - 256/81 </li></ul><ul><li>India - 339/108 ≈ 3.139 </li></ul><ul><li>Around 480, the Chinese mathematician Zu Chongzhi gave the approximation π = 355/113 ≈ 3.141592. </li></ul><ul><li>Around 1400 Madhava (India) calculated Pi correct to 11 decimal places using infinite series. </li></ul>
  5. 5. π in computer age <ul><li>Calculating billions of digits is possible using infinite series. </li></ul><ul><li>The digits of π behave as Pseudo-Random numbers and hence are of interest for Cryptographic Applications. </li></ul><ul><li>The Random number produced by π are of high ‘quality’.However the conjecture that π produces random numbers has still not been proved !! </li></ul>
  6. 6. 1. Madhava of Sangamagrama <ul><li>Calculates the value of π correct up to 14 decimal places in 31 iterations. </li></ul>
  7. 7. Leibniz Formula : <ul><li>Calculates the value of π correct upto 3 decimal places in 10,000 iterations </li></ul>
  8. 8. Euler’s Formula : <ul><li>Calculates the value of π correct upto 3 decimal places in 10,000 iterations. </li></ul>
  9. 9. John Machin : <ul><li>Calculates the value of π correct up to 14 decimal places in 11 iterations. </li></ul>
  10. 10. Ramanujan Formula : <ul><li>Computes the value of correct up to 16 decimal places in 2 iterations. </li></ul><ul><li>Adds 8 decimal places in every iteration. </li></ul>
  11. 11. Bailey–Borwein–Plouffe formula : <ul><li>Calculates the value of π correct up to 15 decimal places in 11 iterations, </li></ul><ul><li>Can be used to calculate any arbitrary digit without calculating the previous digits. </li></ul>
  12. 12. Continued Fraction : <ul><li>Calculates the value of π correct upto 10 decimal places in 10,000 iterations </li></ul>
  13. 13. Infinite Product : <ul><li>Calculates the value of π correct upto 4 decimal places in 10,000 iterations </li></ul>

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