Pi

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Pi

  1. 1. Presented by CLINTA P VARGHESE 9.A
  2. 2. <ul><li>By definition, pi is the ratio of the circumference of a circle to its diameter. Pi is always the same number, no matter which circle you use to compute it. </li></ul>
  3. 5. <ul><li>The area of a circle is pi times the square of the length of the radius, or &quot;pi r squared&quot;: </li></ul><ul><li>A = pi*r^2 </li></ul>
  4. 6. <ul><li>Pi is a very old number. </li></ul><ul><li>We know that the Egyptians and the Babylonians knew about the existence of the constant ratio pi, although they didn't know its value nearly as well as we do today. </li></ul><ul><li>They had figured out that it was a little bigger than 3; the Babylonians had an approximation of 3 1/8 (3.125), and the Egyptians had a somewhat worse approximation of 4*(8/9)^2 (about 3.160484), which is slightly less accurate and much harder to work with. </li></ul><ul><li>modern day technology allows us to calculate pi to billions of decimal places. 3.14 is usually all we need. </li></ul>
  5. 9. <ul><li>Pi is one of the longest numbers ever computed, second only to “e” another IRRATIONAL number with a value of 2.718281828459045 -- </li></ul><ul><li>It never repeats like the decimal values of 1/3=.33333… or 5/7=.7142857142857… </li></ul><ul><li>Visit http://www.sciboard.louisville.edu to see if you can find a computer program that calculates Pi or e. </li></ul>
  6. 10. <ul><li>The early Babylonians and Hebrews used three as a value for Pi. Later, Ahmes, an Egyptian found the area of a circle . Down through the ages, countless people have puzzled over this same question, “What is Pi?&quot; </li></ul><ul><li>From 287 - 212B.C. there </li></ul><ul><li>lived Archimedes, who inscribed </li></ul><ul><li>in a circle and circumscribed </li></ul><ul><li>about a circle, regular polygons. </li></ul><ul><li>The Greeks found Pi to be related to </li></ul><ul><li>cones, ellipses, cylinders and other </li></ul><ul><li>geometric figures. </li></ul>
  7. 11. <ul><li>When mathematicians are faced with quantities which are hard to compute, they try, at least, to pin them betweentwo other quantities which they can compute. </li></ul><ul><li>The Greeks were not able to find any fraction for Pi. </li></ul><ul><li>Today we know that Pi is NOT a rational number and cannot be expressed as a fraction. </li></ul>
  8. 13. Analytical Geometry and Calculus <ul><li>During the 17th century, analytic geometry and calculus were developed. They had a immediate effect on Pi. Pi was freed from the circle! An ellipse has a formula for its area which involves Pi (a fact known by the Greeks); but this is also true of the sphere, cycloid arc, hypoclycloid, the witch, and many other curves. </li></ul>
  9. 14. <ul><li>One of the major chapters in calculus deals with infinite </li></ul><ul><li>series. Several such series have been discovered which </li></ul><ul><li>approximate Pi. These discoveries and the conjuncture </li></ul><ul><li>that Pi is a transcendental number led others to compute </li></ul><ul><li>Pi to more places </li></ul><ul><li>Vega 1794 137 digits </li></ul><ul><li>Dase 1844 201 </li></ul><ul><li>Rutherford 1853 441 </li></ul><ul><li>Shanbis 1873 707 (only 527 were correct) </li></ul>
  10. 15. Anyone for Pi? <ul><li>It’s curious how certain topics in mathematics show up over and over. In the late 1940's two new mathematical streams (electronic computing and statistics) put Pi on the table again. </li></ul>
  11. 16. <ul><li>The development of high speed electronic computing equipment provided a means for rapid computation. Inquiries regarding the number of Pi’s digits </li></ul><ul><li>-- not what the numbers were individually, but how they behave statistically -- provided the motive for additional research. </li></ul>
  12. 17. What is the record today? <ul><li>Around 1950, Borel noted that numbers like the Square Roots of </li></ul><ul><li>2, 3, etc. appear to be a mere jumble of digits, but on the </li></ul><ul><li>average each digit appears a fixed fraction of the </li></ul><ul><li>time. (Some people say this is characteristic of a random set </li></ul><ul><li>of numbers. Do the digits of Pi occur randomly?) </li></ul><ul><li>Such number are called 'normal.' With computers widely </li></ul><ul><li>available the race was on again! </li></ul><ul><li>1950 Eniac in 70 hours produced 2,036 digits </li></ul><ul><li>1954 More in 13 minutes produced 3,093 digits </li></ul><ul><li>1959 IBM 708 in 1 hr 40 min produced 10,000 digits </li></ul><ul><li>1959 Pegasus produced > 100,000 digits </li></ul>
  13. 18. <ul><li>The computation of Pi to 10,000 places may be of no direct scientific usefulness. However, its usefulness in training personnel to use computers and to test such machines appears to be extremely important. Thus the mysterious and wonderful Pi is reduced to a gargle that helps computing machines clear their throats. </li></ul>

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