The Pi - π
Discover the fascinating history and evolution of the mathematical constant pi, from ancient civilizations to modern-day computing records.
When it comes to pi, I think most people are familiar with it. We first encounter it in elementary and junior high school, and now we all understand that pi is the ratio of a circle's circumference to its diameter, approximately equal to 3.14. We also know that pi is both irrational and transcendental.
People's understanding of pi has developed over a long period of time. It took almost 4,000 years from the discovery of pi to determine that it is an irrational number. The earliest historical record of pi can be traced back to the 20th century BC, where an ancient Babylonian stone plaque recorded pi as 25/8 or 3.125. Similarly, an ancient Egyptian artifact known as the Rhind Mathematical Papyrus from the same period showed that the ratio of pi is equal to the square of the fraction 16/9, which approximates to 3.1605. This suggests that the Egyptians had knowledge of pi earlier than other civilizations.
John Taylor, a British writer (1781–1864), noted in his famous book "The Great Pyramid: Why Was It Built, and Who Built It?" that the Pyramid of Khufu, which dates back to around 2500 BC, is linked to pi. Specifically, the ratio of the pyramid's circumference to its height is two times pi, which is the same as the ratio of the circumference to the radius of a circle. Furthermore, the ancient Indian religious masterpiece "Baidao Brahma," written between 800 and 600 BC, demonstrates that pi's ratio is equal to the fraction 339/108, which is roughly 3.139.
It was not until the 3rd century BC that Archimedes, a renowned ancient Greek mathematician and physicist, accurately calculated pi to three decimal places. It took over five hundred years for people to advance the value of π from 3.141 to 3.14159 (achieved by Liu Hui, a Chinese mathematician during the Wei and Jin Dynasties). More than two hundred years later, Zu Chongzhi, a mathematician from the Northern and Southern Dynasties, estimated the value of pi between 3.1415926 and 3.1415927 with a small margin of error. All values of π are precise.
Until the early 15th century, the Arabic mathematician Kasi achieved a breakthrough by obtaining 17 accurate decimal values of pi, surpassing Zu Chongzhi's record of almost a millennium. In 1596, the German mathematician Rudolf Van Coylen calculated the value of π to 20 decimal places and spent the rest of his life calculating it to 35 decimal places.
Since then, the calculation of pi has transitioned from the geometric method period to the analytical method period. During this time, people started using infinite series or infinite continuous products to find π, eliminating the need for complicated calculations involving inscribed and circumscribed polygons. Infinite product expressions, infinite continued fractions, infinite series, and other expressions for the value of π have appeared one after another, resulting in a rapid increase in the accuracy of π calculations. The first fast algorithm was proposed by the British mathematician Machin in 1706, who calculated the value of π to more than 100 decimal places using the following formula:
The value of arctan x can be computed through the Taylor series, which is also known as "Machin's formula." In 1789, a Slovenian mathematician named Jurij Vega established a world record by calculating π up to 140 decimal places, out of which only 137 were correct. This record remained unbeaten for fifty years until 1948, when Ferguson from the UK and Lench from the US jointly calculated π to 808 decimal places, setting a new record for the artificial calculation of π.
The advent of electronic computers revolutionized the calculation of π, leading to significant progress in its computation. In 1949, the first computer in the world, called ENIAC (Electronic Numerical Integrator and Computer), was activated at the Aberdeen Proving Ground in the United States. The following year, Rittweissner, von Neumann, and Metropolis used the ENIAC to compute π up to 2037 decimal places, taking only 70 hours to complete the task.
In 1954, the IBM NORC (Naval Ordnance Research Computer) calculated π to 3089 decimal places in just 13 minutes, setting a new record for the computation of π. With the continual advancement of technology, the computing power of computers has also increased significantly. During the 1960s and 1970s, computer scientists from the US, UK, and France engaged in fierce competition, leading to the calculation of increasingly precise values of π. In 1973, using the CDC 7600 computer, Jean Guilloud and Martin Bouyer discovered the one millionth decimal place of π.
The researchers at Columbia University in the United States calculated pi to 480 million decimal places beyond the decimal point in 1989, and continued computing to reach 1.01 billion digits. In January 2010, French engineer Fabrice Bella extended the calculation to 2.7 trillion decimal places. A few months later, in August 2010, Japanese computer scientist Shigeru Kondo used a home computer and cloud computing to calculate pi to 5 trillion decimal places. The following year, Kondo broke his own record by calculating pi to 10 trillion decimal places beyond the decimal point.
On Pi Day (March 14) of the previous year, Google engineer Emma Iwao achieved a new record by using the Google Computing Engine to calculate pi to 31.4 trillion decimal places beyond the decimal point. Some may question the practical significance of these extremely precise calculations of pi, given that it has already been calculated to more than 30 trillion decimal places beyond the decimal point.
In addition to its well-known use in solving geometric problems involving circles and spheres, pi has numerous applications in other fields. For example, in astronomy, pi is used to calculate the observable range of the universe, and an accuracy of 39 decimal places is sufficient to keep the error below the volume of an atom. In computer information encryption, important documents are often encrypted with completely random numbers generated from pi, providing a high level of security against being cracked. Additionally, pi serves as a ruler for testing computer performance, with greater accuracy leading to stronger performance. It also has significant applications in fields such as trigonometric functions, calculus, alternating current, and radio propagation calculations.
Some scientists even believe that pi is the code of the universe, given its infinite and irregular characteristics similar to those of the universe. Calculating the value of pi may therefore help to unlock the mysteries of the universe.
However, the exact value of pi is not as important as its usefulness as a tool for advancing human civilization. It reflects the evolution of human ideas, concepts, and wisdom, and embodies the spirit of continuous thinking and pursuit.
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