Wednesday, 4 July 2012

History ofThe π

The History of π

In the long history of the number π, there have been many twists and turns, many inconsistencies that reflect the condition of the human race as a whole. Through each major period of world history and in each regional area, the state of intellectual thought, the state of mathematics, and hence the state of π, has been dictated by the same socio-economic and geographic forces as every other aspect of civilization. The following is a brief history, organized by period and region, of the development of our understanding of the number π.

In ancient times, π was discovered independently by the first civilizations to begin agriculture. Their new sedentary life style first freed up time for mathematical pondering, and the need for permanent shelter necessitated the development of basic engineering skills, which in many instances required a knowledge of the relationship between the square and the circle (usually satisfied by finding a reasonable approximation of π). Although there are no surviving records of individual mathematicians from this period, historians today know the values used by some ancient cultures. Here is a sampling of some cultures and the values that they used: Babylonians - 3 1/8, Egyptians - (16/9)^2, Chinese - 3, Hebrews - 3 (implied in the Bible, I Kings vii, 23).

The first record of an individual mathematician taking on the problem of π (often called "squaring the circle," and involving the search for a way to cleanly relate either the area or the circumference of a circle to that of a square) occurred in ancient Greece in the 400's B.C. (this attempt was made by Anaxagoras). Based on this fact, it is not surprising that the Greek culture was the first to truly delve into the possibilities of abstract mathematics. The part of the Greek culture centered in Athens made great leaps in the area of geometry, the first branch of mathematics to be thoroughly explored. Antiphon, an Athenian philosopher, first stated the principle of exhaustion (click on Antiphon for more info). Hippias of Elis created a curve called the quadratrix, which actually allowed the theoretical squaring of the circle, though it was not practical.

In the late Greek period (300's-200's B.C.), after Alexander the Great had spread Greek culture from the western borders of India to the Nile Valley of Egypt, Alexandria, Egypt became the intellectual center of the world. Among the many scholars who worked at the University there, by far the most influential to the history of π was Euclid. Through the publishing of Elements, he provided countless future mathematicians with the tools with which to attack the π problem. The other great thinker of this time, Archimedes, studied in Alexandria but lived his life on the island of Sicily. It was Archimedes who approximated his value of π to about 22/7, which is still a common value today.

Archimedes was killed in 212 B.C. in the Roman conquest of Syracuse. In the years after his death, the Roman Empire gradually gained control of the known world. Despite their other achievements, the Romans are not known for their mathematical achievements. The dark period after the fall of Rome was even worse for π. Little new was discovered about π until well into the decline of the Middle Ages, more than a thousand years after Archimedes' death. (For an example of at least one mediaeval mathematician, see Fibonacci.)

While π activity stagnated in Europe, the situation in other parts of the world was quite different. The Mayan civilization, situated on the Yucatan Peninsula in Central America, was quite advanced for its time. The Mayans were top-notch astronomers, developing a very accurate calendar. In order to do this, it would have been necessary for them to have a fairly good value for π. Though no one knows for sure (nearly all Mayan literature was burned during the Spanish conquest of Mexico), most historians agree that the Mayan value was indeed more accurate than that of the Europeans. The Chinese in the 5th century calculated π to an accuracy not surpassed by Europe until the 1500's. The Chinese, as well as the Hindus, arrived at π in roughly the same method as the Europeans until well into the Renaissance, when Europe finally began to pull ahead.

During the Renaissance period, π activity in Europe began to finally get moving again. Two factors fueled this acceleration: the increasing importance of mathematics for use in navigation, and the infiltration of Arabic numerals, including the zero (indirectly introduced from India) and decimal notation (yes, the great mathematicians of antiquity made all of their discoveries without our standard digits of 0-9!). Leonardo Da Vinci and Nicolas Copernicus made minimal contributions to the π endeavor, but François Viète actually made significant improvements to Archimedes' methods. The efforts of Snellius, Gregory, and John Machin eventually culminated in algebraic formulas for π that allowed rapid calculation, leading to ever more accurate values of π during this period.

In the 1700's the invention of calculus by Sir Isaac Newton and Leibniz rapidly accelerated the calculation and theorization of π. Using advanced mathematics, Leonhard Euler found a formula for π that is the fastest to date. In the late 1700's Lambert (Swiss) and Legendre (French) independently proved that π is irrational. Although Legendre predicted that π is also transcendental, this was not proven until 1882 when Lindemann published a thirteen-page paper proving the validity of Legendre's statement. Also in the 18th century, George Louis Leclerc, Comte de Buffon, discovered an experimental method for calculating π. Pierre Simon Laplace, one of the founders of probability theory, followed up on this in the next century. Click here to learn more about Buffon's and Laplace's method.

Starting in 1949 with the ENIAC computer, digital systems have been calculating π to incredible accuracy throughout the second half of the twentieth century. Whereas ENIAC was able to calculate 2,037 digits, the record as of the date of this article is 206,158,430,000 digits, calculated by researchers at the University of Tokyo. It is highly probable that this record will be broken, and there is little chance that the search for ever more accurate values of π will ever come to an end.

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