Pi (mathematics)
Pi (π) is a mathematical constant that represents the ratio of a circle's circumference to its diameter, approximately equal to 3.14159. This value is irrational, meaning it cannot be expressed as a simple fraction, and its decimal representation is infinite and non-repeating. Throughout history, various cultures have approximated π, with notable contributions from ancient Babylonians, Egyptians, and the Greek mathematician Archimedes, who provided one of the first accurate estimations. Pi's significance extends beyond geometry, appearing in various mathematical contexts and physical laws, including Einstein's equations and the Heisenberg uncertainty principle.
The constant has intrigued mathematicians and enthusiasts alike, leading to competitive memorization challenges and unique experiments to approximate its value. In mathematics, π is also recognized as transcendental, which has profound implications for classical geometric problems. This fascination with π has permeated popular culture, resulting in events like "Pi Day," celebrated on March 14, highlighting its enduring impact on both mathematical and everyday life. As research continues into the properties of π, its digits serve as a benchmark for computer performance and mathematical exploration.
Subject Terms
Pi (mathematics)
Summary: The ratio of a circle’s circumference to it’s diameter, π, is one of the most important constants and the first irrational number encountered by most students.
By definition, pi (π) is the ratio of a circle’s circumference to the diameter. This definition holds for any circle, with the value of π being the constant value 3.14159265358979… This decimal neither terminates nor repeats, making π irrational. Mathematicians and non-mathematicians alike are intrigued by the many appearances of π in diverse situations. Capturing this apparent mysticism in the 1800s, the mathematician Augustus de Morgan wrote, “This mysterious 3.14159. . .which comes in at every door and window, and down every chimney.”
Values Used for Pi
Since the beginning of written mathematics, people have tried to calculate π’s value. Around 2000 BCE, the Babylonians and Egyptians assigned values equal to 3 1/8 (3.125) and 4(8/9)2 (3.1605). In 1100 BCE, the Chinese used π=3, a value which also appears in the Bible (I Kings 5:23). In 300 BCE, Archimedes of Syracuse produced the first “accurate” value, using inscribed and circumscribed 96-sided polygons to produce the approximation 3 10/71 < π < 3 1/7 (or 3.140845. . . < π < 3.142857. . .). Since that time, multiple methods and formulas have been created to determine more exact values of π. Today, powerful computers use similar formulas to calculate values of π to extreme precision, with the current value exceeding 2.7 trillion digits (the record as of January 2010). Two examples of these formulas involving infinite series are
Students in the twenty-first century learn about π in elementary school, and exposure to π continues in later courses in mathematics and physics. Since spherical coordinates are used in many applications, π is found in physical formulas such as Einstein’s field equations, the Heisenberg uncertainty principle, and Coulomb’s law for electric force, which are named after Albert Einstein, Werner Heisenberg, and Charles-Augustin de Coulomb, respectively. Mathematicians and computer scientists describe π as a great stress test for computers because of the seemingly random aspects of its digits.
Algorithms to compute the digits of π are regarded as more important than the digits themselves. Mathematicians continue to investigate other unsolved problems related to π, including attempts to determine how random the digits are.
Applications
The number π has played important roles in multiple situations. In 1767, Johann Lambert proved that π was irrational (it could not be written as the ratio of two integers). Then, in 1882, Ferdinand von Lindemann proved that π was transcendental (it could not be constructed using geometric tools and was not a root of a non-constant polynomial equation with rational coefficients). These two discoveries provided the key to proving the impossibilities of the Greeks’ three problems of antiquity—squaring a circle, trisecting an angle, and duplicating a square.
Considered by many to be a ubiquitous number, π shows up in odd situations. First, in 1777, the naturalist Georges Buffon approximated the value of π experimentally by tossing a needle (length L) on a ruled surface (parallel lines spaced at distance D). If the tossed needle touches a line S times on n tosses, then
Second, the probability that two random integers are relatively prime (they have no common divisor) is
Anyone can try these experiments, either by dropping needles or taking ratios of random integers; many are surprised that both produce good approximations for π. However, complex mathematics is needed to explain “why.”
In 1743, Swiss mathematician Leonhard Euler published the formula eixcos(x)+i sin(x), linking exponentials, trigonometric functions, and complex numbers. Substituting x=π, the result becomes the most beautiful formula in mathematics: eiπ+1=0.
Popular Culture
The fascination with decimal expressions of π has led to competitive memorization contests. The Guinness World Records officially recognized Rajveer Meena as the official record holder in 2015 with 70,000 digits memorized, though others have claimed even more digits. Some people use piems (mnemonic poems); for example, “How I need a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.” In this piem, replace each word with its number of letters, producing π˘3.14159265358979. Hideaki Tomoyori, who held the world record of 40,000 digits memorized from 1987–1995, used a pictorial mnemonic system and explained, “I want to go on with the challenge of memorizing π, for just the same reason that people climb high mountains. I think it’s a wonderful thing to challenge the limits of what we can do… the more one memorizes of it, the closer one comes to the real value of the circle—closer to perfection.” Researchers compared his cognitive abilities with a control group and concluded that he was not superior; they attributed his achievement to extensive practice.
The number π also is connected to some odd events. In 1897, the Indiana State Legislature almost passed a mathematically incorrect bill relating to π and squaring the circle. By its definition, the value of π changes if the circle shifts out of the Euclidean world. That is, in taxicab geometry, or metric geometry on a rectangular lattice structure, the value of π is 4.
The number π is an amazing number, both in its interesting properties and the obsessive attention given it by both mathematicians and non-mathematicians. How else could one explain why on March 14 at 1:59, many people shout, “Happy Pi Day!”
Bibliography
Adrian, Y. E. O. The Pleasures of Pi, e and Other Interesting Numbers. Singapore: World Scientific Publishing, 2006.
Beckmann, Petr. A History of π (Pi). New York: Barnes & Noble, 1971.
Berggren, Lennart, Jon Borwein, and Peter Borwein. Pi: A Source Book. New York: Springer-Verlag, 1997.
Blatner, David. The Joy of π. New York: Walker & Co., 1997.
Horn, Elaine J. "What Is Pi?" LiveScience, 8 Mar. 2022, www.livescience.com/29197-what-is-pi.html. Accessed 9 Mar. 2022.
"Most Pi Places Memorised." Guinness World Records, www.guinnessworldrecords.com/world-records/most-pi-places-memorised. Accessed 9 Mar. 2022.
Takahashi, Masanobu, et al. “One% Ability and Ninety-Nine% Perspiration: A Study of a Japanese Memorist.” Journal of Experimental Psychology. Learning, Memory, and Cognition 32, no. 5 (2006).