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

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

Pi becomes theoretical It appears to have been Archimedes who was the first to obtain a theoretical calculation of pi. 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 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. 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 It has become one of the most prominent missions of the scientific community to calculate pi more and more precisely

Archimedes

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

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

The art of calculating Pi evolves Complex formulas are developed in the European Renaissance to calculate pi. 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. 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. This task is much like Napier’s when he decided to determine the value for logarithms.

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

Detailed Chronology of the Calculation of pi and.ac.uk/~history/HistTopics/Pi_ chronology.html and.ac.uk/~history/HistTopics/Pi_ chronology.html

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

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

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

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.

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

Pi up to 2000 places

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): 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): jp/ jp/

Source Used O’Connor, J. J. and E. F. Robertson. “A History of Pi.” Aug University of St. Andrews. 27 Apr