Story of Pi
On the occasion of Pi Day (March 14 or 3/14), I have written an article on the marvelous history of Pi.
Introduction
Pi (π), the 16th letter of the Greek alphabet, is used to represent the most widely known mathematical constant. Pi equals the circumference of a circle divided by the diameter (π = c/d). No matter how large or small a circle is, pi will always be the same number. Pi is an irrational number, which means that it is a real number with non-repeating decimal expansion. It cannot be represented by an integer ratio and goes on forever, otherwise known as an infinite decimal. There is no exact value, seeing as the number does not end. Many mathematicians and math fans are interested in calculating pi to as many digits as possible. The Guinness World Record for reciting the most digits of pi belongs to Lu Chao of China, who has recited pi to more than 67,000 decimal places.The Pi Search Page website has calculated it with the help of a computer program to 200 million digits.\
History
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 is represented 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. Mathematicians 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. 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.
Indians were the first to observe that the perimeter (circumference) of a circle increases in proportion to its diameter. Therefore, our ancestors established the relation- perimeter / diameter = constant. They didn’t call it Pi though. Since the Indus Valley script is not deciphered, it will be incorrect to claim that π was known in the subcontinent in 3000 BC. But they did know the value of Pi by the time Rigveda was written. The Vedangas and Sulabasutras also mention the value of π. The oldest of them, the Baudhayayana Sulabasutra claims that the perimeter of a pit is 3 times its diameter- therefore approximating the value of π at 3. Many other texts, including the Mahabharata and many Puranas approximate π at the value of 3.
Later, many other Sulabasutras mention the value of π to be 18 * (3–2 √2) = 3.088. The Manava Sulabasutra approximates the value of π to be 28/5= 3.125. The ancient Jaina school of mathematics preferred the approximation π = √10. This value of π has been used not only by Jainas, but also by the greats like Varahamihira, Brahmagupta and Sridhara. Aryabhatta approximated π = 62832/20000 = 3.1416. This was astonishingly correct to 4 decimal places (better than 22/7, which is correct only to 2 places). The Indian values of π (√10, 62832/20000) were later included in Chinese and Arab literature.
Nearly 4000 years ago, Babylonians and Egyptians found that pi was slightly greater than 3, and came up with the value 3 1/8 or 3.125 . 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”. 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”. This implies that pi = 3. 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… . 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, that there would be so many sides that the polygon …would be a circle” . Later, Bryson also calculated the area of polygons circumscribing the circle. This was most likely the first time that a mathematical 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. 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 different 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.
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 in the next several centuries. The earliest value of pi used in China was 3. In 263 AD, Liu 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. 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.
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 leap 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. 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. 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.
Madhava is sometimes called the greatest mathematician-astronomer of medieval India. He came from the town of Sangamagrama in Kerala, near the southern tip of India, and founded the Kerala School of Astronomy and Mathematics in the late 14th Century. Although almost all of Madhava’s original work is lost, he is referred to in the work of later Kerala mathematicians as the source for several infinite series expansions (including the sine, cosine, tangent and arctangent functions and the value of π), representing the first steps from the traditional finite processes of algebra to considerations of the infinite, with its implications for the future development of calculus and mathematical analysis. Unlike most previous cultures, which had been rather nervous about the concept of infinity, Madhava was more than happy to play around with infinity, particularly infinite series. He showed how, although one can be approximated by adding a half plus a quarter plus an eighth plus a sixteenth, etc, (as even the ancient Egyptians and Greeks had known), the exact total of one can only be achieved by adding up infinitely many fractions. But Madhava went further and linked the idea of an infinite series with geometry and trigonometry. He realized that, by successively adding and subtracting different odd number fractions to infinity, he could home in on an exact formula for π (this was two centuries before Leibniz was to come to the same conclusion in Europe). Through his application of this series, Madhava obtained a value for π correct to an astonishing 13 decimal places. Madhava’s use of infinite series to approximate a range of trigonometric functions, which were further developed by his successors at the Kerala School, effectively laid the foundations for the later development of calculus and analysis, and either he or his disciples developed an early form of integration for simple functions. Some historians have suggested that Madhava’s work, through the writings of the Kerala School, may have been transmitted to Europe via Jesuit missionaries and traders who were active around the ancient port of Cochin (Kochi) at the time, and may have had an influence on later European developments in calculus.
After this, little progress was made until a pi explosion in the end of the 16th century. Françleois Viéte, a French lawyer and amateur 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. Unfortunately, his formula is not too useful in calculating because it requires too many iterations before convergence, and the square roots become quite complicated. Even he did not even use his own formula in his calculation of pi. 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. Just three years later, a German named Ludolph Van Ceulen presented 20 digits, using the Archimedean 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.
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. In 1706, William Jones used the symbol pi to represent the ratio. It was not immediately embraced, until 1737, when Leonhard Euler began using the symbol pi; then it was quickly accepted.
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…). Wallis showed his formula to Lord 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 +…))))).
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 ((pi/ 4) = 1, the formula could be used to find pi. Thus, one of the most famous formulas for calculating pi was realized: (pi/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. To compute 100 digits, “you would have to calculate more terms than there are particles in the universe”. 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) -…. Using only six terms of this formula, one can calculate pi = 3.141309, which isn’t too far from the real value. Surely, 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.
In fact, Machin took the initiative to calculate pi with his new formula, and computed 100 places by hand. 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.
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. 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. His argument was, in its simplest form, 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. Some people felt that his proof was not rigorous enough, but in 1794, Adrien Marie Legendre gave another proof that satisfied everyone.
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. 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’ calculation 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. One and a half years later, Levi Smith and John Wrench hit the 1000- digit-mark . 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 functional, and a group of mathematicians fed in punch cards and let the gigantic machine calculate 2037 digits in just seventy hours. Whereas it took Shanks several years to come up with his 707 digits, and Ferguson needed about one year to get 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. 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 algorithm that doubles the number of accurate digits with each iteration, as opposed to previous formulas that only added a handful of digits per calculation. 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 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. 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 radius of a hydrogen atom. 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 what 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 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”. 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?
History of the symbol
The history of the constant ratio of the circumference to the diameter of any circle is as old as man’s desire to measure; whereas the symbol for this ratio known today as π (pi) dates from the early 18th century. Even though it is widely believed that the great Swiss-born mathematician Leonhard Euler (1707–83) introduced the symbol π into common use, in fact it was first used in print in its modern sense in 1706 a year before Euler’s birth by a self-taught mathematics teacher William Jones (1675–1749) in his second book Synopsis Palmariorum Matheseos, or A New Introduction to the Mathematics based on his teaching notes. William Jones used the symbol to represent the platonic concept of pi, an ideal that in numerical terms can be approached, but never reached. Before this, the ratio had been awkwardly referred to in medieval Latin as: quantitas in quam cum multiflicetur diameter, proveniet circumferencia (the quantity which, when the diameter is multiplied by it, yields the circumference).
Before the appearance of the symbol π, approximations such as 22/7 and 355/113 had also been used to express the ratio, which may have given the impression that it was a rational number. Though he did not prove it, Jones believed that π was an irrational number: an infinite, non repeating sequence of digits that could never totally be expressed in numerical form. In Synopsis he wrote: ‘… the exact proportion between the diameter and the circumference can never be expressed in numbers…’. Consequently, a symbol was required to represent an ideal that can be approached but never reached. For this Jones recognised that only a pure platonic symbol would suffice. The symbol π had been used in the previous century in a significantly different way by the rector and mathematician, William Oughtred (c. 1575–1 660), in his book Clavis Mathematicae (first
published in 1631). Oughtred used π to represent the circumference of a given circle, so that his π varied according to the circle’s diameter, rather than representing the constant we know today. The circumference of a circle was known in those days as the ‘periphery’, hence the Greek equivalent ‘π’ of our letter ‘π’.
How Pi is calculated
Pi, which is written as the Greek letter for p, is the ratio of the circumference of any circle to the diameter of that circle. Regardless of the circle’s size, this ratio will always equal pi. In decimal form, the value of pi is approximately 3.14. But pi is an irrational number, meaning that its decimal form neither ends (like 1/4 = 0.25) nor becomes repetitive (like 1/6 = 0.166666…). (To only 18 decimal places, pi is 3.141592653589793238.) Hence, it is useful to have shorthand for this ratio of circumference to diameter.
To calculate pi, try a brief experiment: using a compass, draw a circle. Take one piece of string and place it on top of the circle, exactly once around. Now straighten out the string; its length is called the circumference of the circle. Measure the circumference with a ruler. Next, measure the diameter of the circle, which is the length from any point on the circle straight through its center to another point on the opposite side. (The diameter is twice the radius, the length from any point on the circle to its center.) If you divide the circumference of the circle by the diameter, you will get approximately 3.14 — no matter what size circle you drew! A larger circle will have a larger circumference and a larger radius, but the ratio will always be the same. If you could measure and divide perfectly, you would get 3.141592653589793238…, or pi.
Otherwise said, if you cut several pieces of string equal in length to the diameter, you will need a little more than three of them to cover the circumference of the circle. Pi is most commonly used in certain computations regarding circles. Pi not only relates circumference and diameter. Amazingly, it also connects the diameter or radius of a circle with the area of that circle by the formula: the area is equal to pi times the radius squared.
The value of Pi
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117067982148086513282306647093844609550582231725359408128481117450284102701938521105559644622948954930381964428810975665933446128475648233786783165271201909145648566923460348610454326648213393607260249141273724587006606315588174881520920962829254091715364367892590360011330530548820466521384146951941511609433057270365759591953092186117381932611793105118548074462379962749567351885752724891227938183011949129833673362440656643086021394946395224737190702179860943702770539217176293176752384674818467669405132000568127145263560827785771342757789609173637178721468440901224953430146549585371050792279689258923 542019956112129021960864034418159813629774771309960518707211349999998372978049951059731732816096318595024459455346908302642522308253344685035261931188171010003137838752886587533208381420617177669147303598253490428755468731159562863882353787593751957781857780532171226806613001927876611195909216420198938095257201065485863278865936153381827968230301952035301852968995773622599413891249721775283479131515574857242454150695950829533116861727855889075098381754637464939319255060400927701671139009848824012858361603563707660104710181942955596198946767837449448255379774726847104047534646208046684259069491293313677028989152104752162056966024058038150193511253382430035587640247496473263914199272604269922796782354781636009341721641219924586315030286182974555706749838505494588586926995690927210797509302955321165344987202755960236480665499119881834797753566369807426542527862551818417574672890977772793800081647060016145249192173217214772350141441973568548161361157352552133475741849468438523323907394143334547762416862518983569485562099219222184272550254256887671790494601653466804988627232791786085784383827967976681454100953883786360950680064225125205117392984896084128488626945604241965285022210661186306744278622039194945047123713786960956364371917287467764657573962413890865832645995813390478027590099465764078951269468398352595709825822620522489407726719478268482601476990902640136394437455305068203496252451749399651431429809190659250937221696461515709858387410597885959772975498930161753928468138268683868942774155991855925245953959431049972524680845987273644695848653836736222626099124608051243884390451244136549762780797715691435997700129616089441694868555848406353422072225828488648158456028506016842739452267467678895252138522549954666727823986456596116354886230577456498035593634568174324112515076069479451096596094025228879710893145669136867228748940560101503308617928680920874760917824938589009714909675985261365549781893129784821682998948722658804857564014270477555132379641451523746234364542858444795265867821051141354735739523113427166102135969536231442952484937187110145765403590279934403742007310578539062198387447808478489683321445713868751943506430218453191048481005370614680674919278191197939952061419663428754440643745123718192179998391015919561814675142691239748940907186494231961567945208095146550225231603881930142093762137855956638937787083039069792077346722182562599661501421503068038447734549202605414665925201497442850732518666002132434088190710486331734649651453905796268561005508106658796998163574736384052571459102897064140110971206280439039759515677157700420337869936007230558763176359421873125147120532928191826186125867….and so on.
Conclusion
Pi has captured the imagination of mathematicians for thousands of years and it continues to interest all of us today. Even though it is a mathematical constant, we could never know its exact value as it has infinite decimal places. This fact will surely keep it interesting for thousands of years into the future.
