The History of Pi

David Wilson

History of Mathematics

Rutgers, Spring 2000

Throughout the history of mathematics, one of the most enduring challenges has been the calculation of the ratio between a circle's circumference and diameter, which has come to be known by the Greek letter pi. From ancient Babylonia to the Middle Ages in Europe to the present day of supercomputers, mathematicians have been striving to calculate the mysterious number. They have searched for exact fractions, formulas, and, more recently, patterns in the long string of numbers starting with 3.14159 2653..., which is generally shortened to 3.14. William L. Schaaf once said, "Probably no symbol in mathematics has evoked as much mystery, romanticism, misconception and human interest as the number pi" (Blatner, 1). We will probably never know who first discovered that the ratio between a circle's circumference and diameter is constant, nor will we ever know who first tried to calculate this ratio. The people who initiated the hunt for pi were the Babylonians and Egyptians, nearly 4000 years ago. It is not clear how they found their approximation for pi, but one source (Beckman) makes the claim that they simply made a big circle, and then measured the circumference and diameter with a piece of rope. They used this method to find that pi was slightly greater than 3, and came up with the value 3 1/8 or 3.125 (Beckmann, 11). However, this theory is probably a fantasy based on a misinterpretation of the Greek word "Harpedonaptae," which Democritus once mentioned in a letter to a colleague. The word literally means "rope-stretchers" or "rope-fasteners." The misinterpretation is that these men were stretching ropes in order to calculate circles, while they were actually making measurements in order to mark the property limits and areas for temples, according to (Heath, 121).

A famous Egyptian piece of papyrus gives us another ancient estimation for pi. Dated around 1650 BC, the Rhind Papyrus was written by a scribe named Ahmes. Ahmes wrote, "Cut off 1/9 of a diameter and construct a square upon the remainder; this has the same area as the circle" (Blatner, 8). In other words, he implied that pi = 4(8/9)2 = 3.16049, which is also fairly accurate. Word of this did not spread to the East, however, as the Chinese used the inaccurate value pi = 3 hundreds of years later.

Chronologically, the next approximation of pi is found in the Old Testament. A fairly well known verse, 1 Kings 7:23, says: "Also he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about" (Blatner, 13). This implies that pi = 3. Debates have raged on for centuries about this verse. According to some it was just a simple approximation, while others say that "... the diameter perhaps was measured from outside, while the circumference was measured from inside" (Tsaban, 76). However, most mathematicians and scientists neglect a far more accurate approximation for pi that lies deep within the mathematical "code" of the Hebrew language. In Hebrew, each letter equals a certain number, and a word's "value" is equal to the sum of its letters. Interestingly enough, in 1 Kings 7:23, the word "line" is written Kuf Vov Heh, but the Heh does not need to be there, and is not pronounced. With the extra letter , the word has a value of 111, but without it, the value is 106. (Kuf=100, Vov=6, Heh=5). The ratio of pi to 3 is very close to the ratio of 111 to 106. In other words, pi/3 = 111/106 approximately; solving for pi, we find pi = 3.1415094... (Tsaban, 78). This figure is far more accurate than any other value that had been calculated up to that point, and would hold the record for the greatest number of correct digits for several hundred years afterwards. Unfortunately, this little mathematical gem is practically a secret, as compared to the better known pi = 3 approximation.

When the Greeks took up the problem, they took two revolutionary steps to find pi. Antiphon and Bryson of Heraclea came up with the innovative idea of inscribing a polygon inside a circle, finding its area, and doubling the sides over and over . "Sooner or later (they figured), ...[there would be] so many sides that the polygon ...[would] be a circle" (Blatner, 16). Later, Bryson also calculated the area of polygons circumscribing the circle. This was most likely the first time that a mathem atical result was determined through the use of upper and lower bounds. Unfortunately, the work boiled down to finding the areas of hundreds of tiny triangles, which was very complicated, so their work only resulted in a few digits. (Blatner, 16) At ap proximately the same time, Anaxagoras of Clazomenae started working on a problem that would not be conclusively solved for over 2000 years. After imprisonment for unlawful preaching, Anaxagoras passed his time attempting to square the circle. Cajori wri tes: "This is the first time, in the history of mathematics, that we find mention of the famous problem of the quadrature of the circle, the rock that upon which so many reputations have been destroyed.... Anaxagoras did not offer any solution of it, and seems to have luckily escaped paralogisms" (Cajori 17). Since that time, dozens of mathematicians would rack their brains trying to find a way to draw a square with equal area to a given circle; some would maintain that they had found methods to solve the problem, while others would argue that it was impossible. The problem was finally laid to rest in the nineteenth century.

The first man to really make an impact in the calculation of pi was the Greek, Archimedes of Syracuse. Where Antiphon and Bryson left off with their inscribed and circumscribed polygons, Archimedes took up the challenge. However, he used a slightly dif ferent method than they used. Archimedes focused on the polygons' perimeters as opposed to their areas, so that he approximated the circle's circumference instead of the area. He started with an inscribed and a circumscribed hexagon, then doubled the si des four times to finish with two 96-sided polygons. (Archimedes, 92) His method was as follows...

Given a circle with radius, r = 1, circumscribe a regular polygon A with K = 3(2n-1 sides and semiperimeter an and inscribe a regular polygon B with K = 3(2n-1 sides and semiperimeter bn. This results in a decreasing sequence a1, a2, a3... and an increa sing sequence b1, b2, b3... with each sequence approaching pi. We can use trigonometric notation (which Archimedes did not have) to find the two semiperimeters, which are: an = K tan ((/K) and bn = K sin ((/K). Also: an+1 = 2K tan ((/2K) and bn+1 = 2K si n ((/2K). Archimedes began with a1 = 3 tan ((/3) = 3(3 and b1 = 3 sin ((/3) = 3(3/2 and used 265/153 < (3 < 1351/780. He calculated up to a6 and b6 and finally reached the conclusion that 3 10/71 < b6 < pi < a6 < 3 1/7. Archimedes ended with a 96-sided polygon, and numerous delicate calculations. (Archimedes, 95) The fact that he was able to go that far and derive such a good estimation of pi is a "stupendous feat both of imagination and calculation" (O'Connor, 2).

For the next few hundred years, no significant breakthroughs were made in the search for pi. Gradually, "the lead... passed from Europe to the East" (O'Connor, 3) in the next several centuries. The earliest value of pi used in China was 3. In 263 AD, L iu Hui independently discovered the method used by Bryson and Antiphon, and calculated the perimeters of regular inscribed polygons from 12 up to 192 sides, and arrived at the value pi = 3.14159, which is absolutely correct as far as the first five digits go. Near the end of the 5th century, Tsu Ch'ung-chih and his son Tsu Keng-chih came up with astonishing results, when they calculated 3.1415926 < pi < 3.1415927. The father and son duo used inscribed polygons with as many as 24,576 sides. (Blatner, 25) Soon after, the Hindu mathematician Aryabhata gave the 'accurate' value 62,832/20,000 = 3.1416 (as opposed to Archimedes' 'inaccurate' 22/7 which was frequently used), but he apparently never used it, nor did anyone else for several centuries. (Beckmann, 24) Another Indian mathematician, Brahmagupta, took a novel approach. He calculated the perimeters of inscribed polygons with 12, 24, 48, and 96 sides as (9.65, (9.81, (9.86, and (9.87 respectively. "And then, armed with this information, he made the l eap of faith that as the polygons approached the circle, the perimeter, and therefore pi, would approach the square root of 10 [=3.162...]. He was, of course, quite wrong" (Blatner, 26). Although this is not as accurate as other values that had already been calculated, it gained quite a bit of popularity as an approximation for pi for at least a few hundred years. "Maybe because the square root of 10 is so easy to convey and remember, this was the value that... spread from India to Europe and was used by mathematicians... throughout the Middle Ages" (Blatner, 26). By the 9th century, mathematics and science prospered in the Arab cultures. It is unclear whether the Arabian mathematician, Mohammed ibn Musa al'Khwarizmi, attempted to calculate pi, but it is clear which values he used. He used the approximations 3 1/ 7, the square root of 10, and 62,832/20,000. Strangely, though, the last and most accurate value was seemingly forgotten by the Arabs and replaced by less accurate values. (Cajori, 104)

After this, little progress was made until a pi explosion in the end of the 16th century. Françle;ois Viéte, a French lawyer and amateur (but great) mathematician, used Archimedes' method, starting with two hexagons and doubling the number of sides sixteen times, to finish with 393,216 sides. His final result was that 3.1415926535 < pi < 3.1415926537. More importantly, though, Viéte became the first man in history to describe pi using an infinite product. His formula was: 2/pi = ((1/2)(((1/2 + 1/2 ((1/2))(((1/2 + 1/2((1/2 + 1/2(1/2))pi.... Unfortunately, this equation is not too useful in calculating ( because it requires too many iterations before convergence, and the square roots become quite complicated. He did not even use his own formula in his calculation of pi. (Beckmann, 92) Still, it was an innovative discovery that would open many doors in the future. In 1593, Adrianus Romanus used a circumscribed polygon with 230 sides to compute pi to 17 digits after the decimal, of which 15 were correct. (O'Connor, 3) Just three years later, a German named Ludolph Van Ceulen presented 20 digits, using the Archimede an method with polygons with over 500 million sides. Van Ceulen spent a great part of his life hunting for pi, and by the time he died in 1610, he had accurately found 35 digits. His accomplishments were considered so extraordinary that the digits were cut into his tombstone in St. Peter's Churchyard in Leyden. Still today, Germans refer to pi as the Ludolphian Number to honor the man who had such great perseverance. (Cajori, 143) It should be noted that up to this point, there was no symbol to signify the ratio of a circle's circumference to its diameter. This changed in 1647 when William Oughtred published Clavis Mathematicae and used (/( to denote the ratio. It was not immediately embraced, until 1737, when Leonhard Euler began using the symbol pi; then it was quickly accepted. (Cajori, 158) In 1650, John Wallis used a very complicated method to find another formula for pi. Basically, he approximated the area of a quarter circle using infinitely small rectangles, and arrived at the formula 4/pi = (3(3(5(5(7(7(9...)/(2(4(4(6(6(8(8...) which is usually simplified to pi/2 = (2(2(4(4(6(6(8(8...)/(1(3(3(5(5(7(7(9...). One source describes his method as "extremely difficult and complicated" (Berggren, 292) while another source says it is "remarkable" (Cajori, 186). Wallis showed his formula to Lor d Brouncker, the president of the Royal Society, who turned it into a continued fraction: pi = 4/(1 + 1/(2 + 9/(2 + 25/(2 + 49/(2 +...))))). (Cajori, 188)

In 1672, James Gregory wrote about a formula that can be used to calculate the angle given the tangent for angles up to 45pi. The formula is: arctan (t) = t - t3/3 + t5/5 -t7/7 + t9/9.... Ten years later, Gottfried Leibniz pointed out that since tan ((/ 4) = 1, the formula could be used to find pi. (Berggren, 92) Thus, one of the most famous formulas for calculating pi was realized: (/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9.... This elegant formula is one of the simplest ever discovered to calculate pi, but it is also fairly useless; 300 terms of the series are required to get only 2 decimal places, and 10,000 terms are required for 4 decimal places. (O'Connor, 3) To compute 100 digits, "you would have to calculate more terms than there are particles in the univ erse" (Blatner, 42). However, this formula set the stage for a handful of other formulas that would be more effective. For example, using the knowledge that arctan (1/(3) = (/6, you can derive the following equation: arctan (1/(3) = (/6 = 1/(3 - 1/(3(3( 3) + 1/(9(3(5) - .... After some algebra, it simplifies to: (/6 = (1/(3)(1 - 1/(3(3) + 1/(5(32) - 1/(7(33) + 1/(9(34) -.... (O'Connor, 4) Using only six terms of this formula, one can calculate pi = 3.141309, which isn't too far from the real value. Sur ely, the 17th-century mathematicians were onto something. It was just a matter of time until they discovered a formula that was even better.

The world didn't have to wait too long, after all, before another formula was discovered. In 1706, John Machin, a professor of astronomy in London, armed with the knowledge that arctan x + arctan y = arctan (x+y)/(1-xy), discovered the wonderful formula : pi/4 = 4 arctan (1/5) - arctan (1/239) = 4(1/5 - 1/(3(53) + 1/(5(55) - ...) - (1/239 - 1/(3(2393) + 1/(5(2395) - ...). The reason that this formula is such an improvement over the previous one is that the number 239 is so large that we do not need very many terms of arctan (1/239) before it converges. The other term, arctan (1/5) involves easy computations when computing terms by hand, since it involves finding reciprocals of powers of 5. (Blatner, 43) In fact, Machin took the initiative to calculate p i with his new formula, and computed 100 places by hand. (Cajori, 206) Over the next 150 years, several men used the exact same formula to find more and more digits. In 1873, an Englishman named William Shanks used the formula to calculate 707 places of pi. Many years later, it was discovered that somewhere along the line, Shanks had omitted two terms, with the result that only the first 527 digits were correct. (Berggren, 627)

"By 1750, the number pi had been expressed by infinite series,... its value had been computed [to over 100 digits]... and it had been given its present symbol. All these efforts, however, had not contributed to the solution of the ancient problem of the quadrature of the circle" (Struik, 369). The first step was taken by the Swiss mathematician Johann Heinrich Lambert when he proved the irrationality of pi first in 1761 and then in more detail in 1767. (Struik, 369) His argument was, in its simplest for m, that if x is a rational number, then tan x cannot be rational; since tan pi/4 = 1, pi/4 cannot be rational, and therefore pi is irrational. (Cajori, 246) Some people felt that his proof was not rigorous enough, but in 1794, Adrien Marie Legendre gave ano ther proof that satisfied everyone. Furthermore, Legendre also gave the first proof that (2 is irrational. (Berggren, 297)

For the next hundred years, no major events occurred in the pursuit of pi. More and more digits were computed, but there were no earth-shattering breakthroughs. In 1882, Ferdinand von Lindemann proved the transcendence of pi. (Berggren, 407) Since this means that pi is not a solution of any algebraic equation, it lay to rest the uncertainty about squaring the circle. Finally, after literally thousands and thousands of lifetimes of mental toil and strain, mathematicians finally had an absolute answer that the circle could not be squared. Nonetheless, there are still some amateur mathematicians today who do not understand the significance of this result, and futilely look for techniques to square the circle.

In the twentieth century, computers took over the reigns of calculation, and this allowed mathematicians to exceed their previous records to get to previously incomprehensible results. In 1945, D. F. Ferguson discovered the error in William Shanks' calc ulation from the 528th digit onward. Two years later, Ferguson presented his results after an entire year of calculations, which resulted in 808 digits of pi. (Berggren, 406) One and a half years later, Levi Smith and John Wrench hit the 1000-digit-mark . (Berggren, 685) Finally, in 1949, another breakthrough emerged, but it was not mathematical in nature; it was the speed with which the calculations could be done. The ENIAC (Electronic Numerical Integrator and Computer) was finally completed and funct ional, and a group of mathematicians fed in punch cards and let the gigantic machine calculate 2037 digits in just seventy hours. (Beckmann, 180) Whereas it took Shanks several years to come up with his 707 digits, and Ferguson needed about one year to g et 808 digits, the ENIAC computed over 2000 digits in less than three days!

"With the advent of the electronic computer, there was no stopping the pi busters" (Blatner, 51). John Wrench and Daniel Shanks found 100,000 digits in 1961, and the one-million-mark was surpassed in 1973. In 1976, Eugene Salamin discovered an algorith m that doubles the number of accurate digits with each iteration, as opposed to previous formulas that only added a handful of digits per calculation. (Blatner, 52) Since the discovery of that algorithm, the digits of pi have been rolling in with no end in sight. Over the past twenty years, six men in particular, including two sets of brothers, have led the race: Yoshiaki Tamura, Dr. Yasumasa Kanada, Jonathan and Peter Borwein, and David and Gregory Chudnovsky. Kanada and Tamura worked together on many pi projects, and led the way throughout the 1980s, until the Chudnovskys broke the one-billion-barrier in August 1989. In 1997, Kanada and Takahashi calculated 51.5 billion (3(234) digits in just over 29 hours, at an average rate of nearly 500,000 digits per second! The current record, set in 1999 by Kanada and Takahashi, is 68,719,470,000 digits. (Blatner, 59) There is no knowing where or when the search for pi will end. Certainly, the continued calculations are unnecessary. Just thirty-nine decimal places would be enough to compute the circumference of a circle surrounding the known universe to within the ra dius of a hydrogen atom. (Berggren, 656) Surely, there is no conceivable need for billions of digits. At the present time, the only tangible application for all those digits is to test computers and computer chips for bugs. But digits aren't really wha t mathematicians are looking for anymore. As the Chudnovsky brothers once said: "We are looking for the appearance of some rules that will distinguish the digits of pi from other numbers. If you see a Russian sentence that extends for a whole page, with hardly a comma, it is definitely Tolstoy. If someone gave you a million digits from somewhere in pi, could you tell it was from pi? We don't really look for patterns; we look for rules" (Blatner, 68). Unfortunately, the Chudnovskys have also said that no other calculated number comes closer to a random sequence of digits. Who knows what the future will hold for the almost magical number pi?

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Various Formulas for Computing Pi

Wallis

p /2=(2.2.4.4.6.6.8.8. ...)/(1.3.3.5.5.7.7.9. ...) Machin

p /4=4 arctan(1/5)-arctan(1/239) Ferguson

p /4= 3 arctan(1/4)+arctan(1/20)+arctan(1985) Euler

p /4= 5 arctan(1/7)+2 arctan(3/79) Euler

p 2/6=1/22+1/32+ 1/42+1/52+ ... Euler

ei p +1=0 Borwein and Borwein

1/ p =12 S [(-1)n(6n)!/(n!)3(3n)!] [(A+nB)/Cn+1/2]; where A=212175710912 Ö (61) +1657145277365;

B=13773980892672 Ö (61) +107578229802750;

C=[5280(236674+30303 Ö (61)]3 Borwein, Bailey, and Plouffe p = S [4/(8n+1)-2/(8n+4)-1/(8n+5)-1/(8n+6)](16)-n This formula enables one to calculate the nth digit of pi, in hexadecimal notation, without being forced to calculate the preceding n-1 digits.

Works Cited