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The History of Pi

The History of Pi. By Joel Chorny Phys 001 Spring 2004. Pi is ancient. “The fact that the ratio of the circumference to the diameter of a circle is constant has been known for so long that it is quite untraceable” (O’Connor).

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The History of Pi

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  1. The History of Pi By Joel Chorny Phys 001 Spring 2004

  2. Pi is ancient • “The fact that the ratio of the circumference to the diameter of a circle is constant has been known for so long that it is quite untraceable” (O’Connor). • The Bible contains a verse that tells us a value of pi that was used. • “And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and its height was five cubits: and a line of thirty cubits did compass it about”- (I Kings 7, 23) • Here the value of pi is given as 3, not very accurate, not even for its time.

  3. Even the Egyptian and Mesopotamian values of 25/8= 3.125 and √10= 3.162 have been traced to much earlier dates than the biblical value of 3 • The earliest values of pi were almost certainly empirically determined, which means they were found by measurement. Rhind Papyrus

  4. Pi becomes theoretical • It appears to have been Archimedes who was the first to obtain a theoretical calculation of pi. • He concluded the following: 223/71<pi<22/7 • Archimedes used inequalities very sophisticatedly here to show that he knew pi did not equal 22/7. He never claimed to have found the exact value. • It has become one of the most prominent missions of the scientific community to calculate pi more and more precisely

  5. Archimedes

  6. Pi becomes more and more exact • Ptolemy calculated pi to be 3.1416 • Zu Chongzhi obtained the value pi= 355/113 • Al-Khwarizmi without knowledge of Ptolemy’s work found pi to be 3.1416 • Al-Kashi calculated pi to 14 decimal places • Roomen calculated pi to 17 decimal places • Van Ceulen calculated pi to 35 decimal places

  7. Al-Khwarizmi • Lived in Baghdad • Gave his name to the word “algorithm” • The word “algebra” comes from al jabr, the title of one of his books • Was the pioneer of the calculation of pi in the East Al-Khwarizmi

  8. The art of calculating Pi evolves • Complex formulas are developed in the European Renaissance to calculate pi. • With these formulas available, the difficulty in calculating pi comes only in the sheer time consumption and boredom of continuing the calculation. • This task is much like Napier’s when he decided to determine the value for logarithms.

  9. Some people were “dedicated” enough to actually spend incredible amounts of time and effort continuing the calculation of pi. • 1699: Sharp gets 71 correct digits • 1701: Machin gets 100 digits • 1719: de Lagny gets 112 correct digits • 1789: Vega gets 126 places • 1794: Vega gets 136 places • 1841: Rutherford gets 152 digits • 1853: Rutherford gets 440 digits • 1873: Shanks calculates 707 places of which 527 were correct

  10. Detailed Chronology of the Calculation of pi http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_chronology.html

  11. Augustus de Morgan • English mathematician born in India • Looked at Shanks’ 707-digit calculation of pi. • Noticed that there was a suspicious shortage of 7s. • In 1945 Ferguson discovers that Shanks had made a mistake in the 528th place, which lead to all the following digits to be wrong. De Morgan

  12. More precision becomes available • Pi was calculated to 2000 places with the use of a computer in 1949. • In this calculation, and all calculations following it, the number of 7s does not differ significantly from its expectation. • The record number of decimal places for pi calculated in 1999 was 206,158,430,000. However, this record has already been broken.

  13. The Notation of pi • The first to use the symbol πwith its current meaning was William Jones in 1706. He was a Welsh mathematician. • Euler adopted the symbol in 1737 and it soon became a standard. William Jones Leonhard Euler

  14. What does all this have to do with us? Throughout the semester we have been learning about how improvements have been made in the art of measurement. Tyco Brahe used instruments the size of buildings to take accurate measurements of the movement of the stars and planets. The constant attempt to improve on our understanding of pi is similarly to be able to make more accurate measurements.

  15. Just as scientists have tried to calculate the speed of light to the most accurate decimal possible, scientists are trying to define pi to the most accurate decimal. It is becoming increasingly often that pi is defined in terms of more decimal places

  16. Pi up to 2000 places • 3.14159265358979323846264338327950288419716939937510582097494459 230781640628620899862803482534211706798214808651328230664709384 460955058223172535940812848111745028410270193852110555964462294 895493038196442881097566593344612847564823378678316527120190914 564856692346034861045432664821339360726024914127372458700660631 558817488152092096282925409171536436789259036001133053054882046 652138414695194151160943305727036575959195309218611738193261179 310511854807446237996274956735188575272489122793818301194912983 367336244065664308602139494639522473719070217986094370277053921 717629317675238467481846766940513200056812714526356082778577134 275778960917363717872146844090122495343014654958537105079227968 925892354201995611212902196086403441815981362977477130996051870 72113499999983729780499510597317328160963185950244594553469083 026425223082533446850352619311881710100031378387528865875332083 814206171776691473035982534904287554687311595628638823537875937 519577818577805321712268066130019278766111959092164201989380952 572010654858632788659361533818279682303019520353018529689957736 225994138912497217752834791315155748572424541506959508295331168 617278558890750983817546374649393192550604009277016711390098488 240128583616035637076601047101819429555961989467678374494482553 797747268471040475346462080466842590694912933136770289891521047 521620569660240580381501935112533824300355876402474964732639141 992726042699227967823547816360093417216412199245863150302861829 745557067498385054945885869269956909272107975093029553211653449 872027559602364806654991198818347977535663698074265425278625518 184175746728909777727938000816470600161452491921732172147723501 414419735685481613611573525521334757418494684385233239073941433 345477624168625189835694855620992192221842725502542568876717904 946016534668049886272327917860857843838279679766814541009538837 863609506800642251252051173929848960841284886269456042419652850 222106611863067442786220391949450471237137869609563643719172874 677646575739624138908658326459958133904780275901

  17. If you want to get a sense of how huge the amount of decimal places calculated for pi is, go to the following url (Load time is pretty long): http://3.141592653589793238462643383279502884197169399375105820974944592.jp/

  18. Source Used O’Connor, J. J. and E. F. Robertson. “A History of Pi.” Aug. 2001. University of St. Andrews. 27 Apr. 2004 <http://www-history.mcs.st-andrews.ac.uk/HistTopics/Pi_through_the_ages.html>.

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