The History of PiThe History of Pi

By Joel ChornyBy Joel Chorny

Phys 001Phys 001

Spring 2004Spring 2004

Pi is ancientPi is ancient

““The fact that the ratio of the circumference to The fact that the ratio of the circumference to the diameter of a circle is constant has been the diameter of a circle is constant has been known for so long that it is quite untraceable” known for so long that it is quite untraceable” (O’Connor).(O’Connor).

The Bible contains a verse that tells us a value The Bible contains a verse that tells us a value of pi that was used.of pi that was used. ““And he made a molten sea, ten cubits from the one And he made a molten sea, ten cubits from the one

brim to the other: it was round all about, and its height brim to the other: it was round all about, and its height was five cubits: and a line of thirty cubits did compass was five cubits: and a line of thirty cubits did compass it about”- (I Kings 7, 23)it about”- (I Kings 7, 23)

Here the value of pi is given as 3, not very accurate, not even Here the value of pi is given as 3, not very accurate, not even for its time.for its time.

Even the Egyptian and Even the Egyptian and Mesopotamian values of Mesopotamian values of 25/8= 3.125 and √10= 25/8= 3.125 and √10= 3.162 have been traced to 3.162 have been traced to much earlier dates than the much earlier dates than the biblical value of 3biblical value of 3

The earliest values of pi The earliest values of pi were almost certainly were almost certainly empirically determined, empirically determined, which means they were which means they were found by measurement.found by measurement.

Rhind Papyrus

Pi becomes theoreticalPi becomes theoretical

It appears to have been Archimedes who was It appears to have been Archimedes who was the first to obtain a theoretical calculation of pi.the first to obtain a theoretical calculation of pi. He concluded the following: 223/71<pi<22/7He concluded the following: 223/71<pi<22/7

Archimedes used inequalities very Archimedes used inequalities very sophisticatedly here to show that he knew pi did sophisticatedly here to show that he knew pi did not equal 22/7. He never claimed to have found not equal 22/7. He never claimed to have found the exact value.the exact value.

It has become one of the most prominent It has become one of the most prominent missions of the scientific community to calculate missions of the scientific community to calculate pi more and more preciselypi more and more precisely

Archimedes

Ptolemy calculated pi to be 3.1416Ptolemy calculated pi to be 3.1416Zu Chongzhi obtained the value pi= 355/113Zu Chongzhi obtained the value pi= 355/113Al-Khwarizmi without knowledge of Ptolemy’s Al-Khwarizmi without knowledge of Ptolemy’s

work found pi to be 3.1416work found pi to be 3.1416Al-Kashi calculated pi to 14 decimal placesAl-Kashi calculated pi to 14 decimal placesRoomen calculated pi to 17 decimal placesRoomen calculated pi to 17 decimal placesVan Ceulen calculated pi to 35 decimal placesVan Ceulen calculated pi to 35 decimal places

Pi becomes more and more exactPi becomes more and more exact

Al-KhwarizmiAl-Khwarizmi

Lived in BaghdadLived in Baghdad Gave his name to the Gave his name to the

word “algorithm”word “algorithm” The word “algebra” The word “algebra”

comes from comes from al jabr, al jabr, the title of one of his the title of one of his booksbooks

Was the pioneer of Was the pioneer of the calculation of pi in the calculation of pi in the Eastthe East

Al-Khwarizmi

The art of calculating Pi evolvesThe art of calculating Pi evolves

Complex formulas are developed in the Complex formulas are developed in the European Renaissance to calculate pi.European Renaissance to calculate pi.

With these formulas available, the difficulty With these formulas available, the difficulty in calculating pi comes only in the sheer in calculating pi comes only in the sheer time consumption and boredom of time consumption and boredom of continuing the calculation. continuing the calculation.

This task is much like Napier’s when he This task is much like Napier’s when he decided to determine the value for decided to determine the value for logarithms.logarithms.

Some people were “dedicated” enough to Some people were “dedicated” enough to actually spend incredible amounts of time actually spend incredible amounts of time and effort continuing the calculation of pi.and effort continuing the calculation of pi.1699: Sharp gets 71 correct digits1699: Sharp gets 71 correct digits1701: Machin gets 100 digits1701: Machin gets 100 digits1719: de Lagny gets 112 correct digits 1719: de Lagny gets 112 correct digits 1789: Vega gets 126 places 1789: Vega gets 126 places 1794: Vega gets 136 places1794: Vega gets 136 places1841: Rutherford gets 152 digits1841: Rutherford gets 152 digits1853: Rutherford gets 440 digits1853: Rutherford gets 440 digits1873: Shanks calculates 707 places of which 1873: Shanks calculates 707 places of which

527 were correct527 were correct

Detailed Chronology of the Detailed Chronology of the Calculation of piCalculation of pi

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_chronology.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Pi_chronology.html

Augustus de MorganAugustus de Morgan English mathematician English mathematician

born in Indiaborn in India Looked at Shanks’ 707-Looked at Shanks’ 707-

digit calculation of pi.digit calculation of pi. Noticed that there was a Noticed that there was a

suspicious shortage of suspicious shortage of 7s. 7s.

In 1945 Ferguson In 1945 Ferguson discovers that Shanks discovers that Shanks had made a mistake in had made a mistake in the 528the 528thth place, which place, which lead to all the following lead to all the following digits to be wrong.digits to be wrong.

De Morgan

More precision becomes availableMore precision becomes available

Pi was calculated to 2000 places with the Pi was calculated to 2000 places with the use of a computer in 1949.use of a computer in 1949.

In this calculation, and all calculations In this calculation, and all calculations following it, the number of 7s does not following it, the number of 7s does not differ significantly from its expectation. differ significantly from its expectation.

The record number of decimal places for The record number of decimal places for pi calculated in 1999 was pi calculated in 1999 was 206,158,430,000. However, this record 206,158,430,000. However, this record has already been broken.has already been broken.

The Notation of piThe Notation of pi

The first to use the The first to use the symbol symbol ππ with its with its current meaning was current meaning was William Jones in William Jones in 1706. He was a 1706. He was a Welsh mathematician.Welsh mathematician.

Euler adopted the Euler adopted the symbol in 1737 and it symbol in 1737 and it soon became a soon became a standard.standard.

William Jones

Leonhard Euler

What does all this have to do with What does all this have to do with us?us?

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

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

3.14159265358979323846264338327950288419716939937510582097494459 3.14159265358979323846264338327950288419716939937510582097494459 230781640628620899862803482534211706798214808651328230664709384 230781640628620899862803482534211706798214808651328230664709384 460955058223172535940812848111745028410270193852110555964462294 460955058223172535940812848111745028410270193852110555964462294 895493038196442881097566593344612847564823378678316527120190914 895493038196442881097566593344612847564823378678316527120190914 564856692346034861045432664821339360726024914127372458700660631 564856692346034861045432664821339360726024914127372458700660631 558817488152092096282925409171536436789259036001133053054882046 558817488152092096282925409171536436789259036001133053054882046 652138414695194151160943305727036575959195309218611738193261179 652138414695194151160943305727036575959195309218611738193261179 310511854807446237996274956735188575272489122793818301194912983 310511854807446237996274956735188575272489122793818301194912983 367336244065664308602139494639522473719070217986094370277053921 367336244065664308602139494639522473719070217986094370277053921 717629317675238467481846766940513200056812714526356082778577134 717629317675238467481846766940513200056812714526356082778577134 275778960917363717872146844090122495343014654958537105079227968 275778960917363717872146844090122495343014654958537105079227968 925892354201995611212902196086403441815981362977477130996051870 925892354201995611212902196086403441815981362977477130996051870 72113472113499999999999983729780499510597317328160963185950244594553469083 83729780499510597317328160963185950244594553469083 026425223082533446850352619311881710100031378387528865875332083 026425223082533446850352619311881710100031378387528865875332083 814206171776691473035982534904287554687311595628638823537875937 814206171776691473035982534904287554687311595628638823537875937 519577818577805321712268066130019278766111959092164201989380952 519577818577805321712268066130019278766111959092164201989380952 572010654858632788659361533818279682303019520353018529689957736 572010654858632788659361533818279682303019520353018529689957736 225994138912497217752834791315155748572424541506959508295331168 225994138912497217752834791315155748572424541506959508295331168 617278558890750983817546374649393192550604009277016711390098488 617278558890750983817546374649393192550604009277016711390098488 240128583616035637076601047101819429555961989467678374494482553 240128583616035637076601047101819429555961989467678374494482553 797747268471040475346462080466842590694912933136770289891521047 797747268471040475346462080466842590694912933136770289891521047 521620569660240580381501935112533824300355876402474964732639141 521620569660240580381501935112533824300355876402474964732639141 992726042699227967823547816360093417216412199245863150302861829 992726042699227967823547816360093417216412199245863150302861829 745557067498385054945885869269956909272107975093029553211653449 745557067498385054945885869269956909272107975093029553211653449 872027559602364806654991198818347977535663698074265425278625518 872027559602364806654991198818347977535663698074265425278625518 184175746728909777727938000816470600161452491921732172147723501 184175746728909777727938000816470600161452491921732172147723501 414419735685481613611573525521334757418494684385233239073941433 414419735685481613611573525521334757418494684385233239073941433 345477624168625189835694855620992192221842725502542568876717904 345477624168625189835694855620992192221842725502542568876717904 946016534668049886272327917860857843838279679766814541009538837 946016534668049886272327917860857843838279679766814541009538837 863609506800642251252051173929848960841284886269456042419652850 863609506800642251252051173929848960841284886269456042419652850 222106611863067442786220391949450471237137869609563643719172874 222106611863067442786220391949450471237137869609563643719172874 677646575739624138908658326459958133904780275901 677646575739624138908658326459958133904780275901

Pi up to 2000 places

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

http://3.141592653589793238462643383279502884197169399375105820974944592.jp/http://3.141592653589793238462643383279502884197169399375105820974944592.jp/

Source Used Source Used

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