The ongoing mystery, irrationality and transcendence of pi

Today is National and World Pi Day because the numbers of the day (3-14) match the first three digits for pi or π, the Greek letter, 3.1415926535897... Although most people think that π is relegated to just geometry and trigonometry, the number pervades all of mathematics and the natural sciences, even statistics.

Several thousand years ago the Egyptians, the Babylonians, the Chinese and the Ancient Greeks tried to make sense of the world through mathematics, an abstract way to envision and explain the operations of Nature, not as the activities of the gods. Over time geometry developed, which could explain much of the world. For example, Euclid and his various axioms were employed to describe much of the natural world. However, when it came to circles and non-linear lines, there remained a mystery among all the Ancients, which was π.

It had long been recognized (and still taught to reluctant students in high school geometry) that the ratio of the circumference of any circle to its diameter is a constant. The Ancients knew this, but the value of that constant eluded them. They realized, however, that there were approximations, e.g., the fractions 25/8, 22/7, 256/81, etc., that were close, and these fractions were employed for centuries as substitutes for pi. 

Over two thousand years ago Archimedes carried this approximation technique to its logical limit, using techniques akin to calculus infinities approaches, and was able to obtain very close estimates of π to whatever tolerance was needed, e.g., through circumscribing and inscribing large numbers of polygons, e.g., an algorithm employing up to 96 such polygons for an accuracy between 3.1408 and 3.14285, about 99.9% accuracy. But, around the year 480 A.D., Chinese mathematician Zu Chongzhi used this approach with 12,288 polygons, and created a far more accurate fractional approximation, 355/113, roughly 99.99999% accurate, which was the best approximation for π for the next 800 years.

As a side note, through recent discoveries, Archimedes is also credited with understanding aspects of calculus long before Newton and Leibnitz, who developed differential and integral calculus just over three hundred years ago. Had the Roman soldier not killed Archimedes in the siege of Syracuse, our world may have been very different. But, I digress.

Clearly, these fractional representations of π were all approximations and not a pure answer, which galled the Ancients at their inability to solve the conundrum. Indeed, the purity in mathematics was at the heart of Euclidian geometry’s goals of solving problems. For example, in their effort to solve the π enigma, the Greeks were famous in their efforts to “square the circle,” i.e., geometrically constructing a square having the same area as a given circle, and asking whether Euclid’s axioms posit the existence of such a number. However, the Greeks and many others later could not do it, which had profound implications to Plato regarding the usefulness of Euclid’s theorems or even mathematics to actually describe the real world. In short, the quest was impossible. But why?

With Euclid and the pre-Socratics trying to explain the world in physical ways, e.g., Democritus postulating atoms in a very logical way 2,500 years ago, it is sad that the mystery of π seems to have derailed the very influential thinkers Socrates and Plato to fully trust mathematics. Accordingly, Plato looked to another realm to describe the world: using his forms or abstractions. For example, the concepts of a circle and π were perfect, idealized forms, but every attempt to depict them in the real world would, by definition, be imperfect. This philosophical view held sway until the Renaissance started new ways of thinking.

But, back to π. We now know that pi is both Irrational and Transcendental. An irrational number is defined as a number that is not a ratio of two whole numbers, i.e., fractions. This irrationality of pi is strongly suggested by Archimedes’ and others’ succession of better and better fractional approximations, without a final answer. Also, with computerization it has been found that the digits of pi have no pattern, and for several trillion digits pass the mathematical test of normality, i.e., all of the digits appear equally often in the series. The irrational nature of pi was formally proven in 1761. 

A transcendent number is defined as a number that is not the root of any non-zero polynomial with rational coefficients, which is a modern way of saying you cannot square the circle.  The transcendence of π was proven in 1882. The staggering notion that the digits go on and on, without repeating or in any pattern to infinity, was (and remains) hard to grasp, the immensity of which was something well understood to Aristotle and others. Over a hundred years ago, however, mathematician George Cantor tackled the mathematical problem of infinity and actually demonstrated the nuances between infinities. π is also computed by various techniques, e.g., equations and trigonometric series, that have terms that go to infinity.

The use of the Greek letter π in this context dates from about three hundred years ago when the great mathematician Leonhard Euler started popularizing it. Mathematician William Jones in 1706 is accredited with being the first to symbolize the circle circumference-to-diameter ratio as π, which is also attributed to the Greek word for perimeter. Prior to computers, pi calculation was a laborious and very error-prone endeavor. With the advent of computing, the mere six or seven hundred digit manual calculations not too many decades ago have jumped to many trillions of digits.

Despite all of the mathematical rigor of the modern era, π remains a mystery, a constant that in a way is inconstant. Of course, there are many other such enigmatic irrational and transcendent numbers out there, e.g., e (2.71828182845…), but π is the oldest of these cosmic constants for us humans. On a related note, this is the 50th anniversary of Stanley Kubrik’s 2001: A Space Odyssey, an inscrutable movie that still contains innumerable mysteries. It is also the 20th anniversary of π, the movie, a psychological thriller about the irrationality of π and the human mind. In Star Trek, Mr. Spock crashed a hostile computer making it calculate pi precisely. π also pops up once and a while in TV shows, such as the Simpsons.

This magical number is everywhere, and is part of our lives – even if you hated high school geometry and math. Indeed, we are all still trying to understand the meanings of π.

Raymond Van Dyke is an intellectual property/patent attorney, educator and a science and technology enthusiast. He has a B.S. in mathematics/computer science and was admitted Pi Mu Epsilon, an honorary mathematics society, has an M.S. in Computer Science, and a J.D. from the University of North Carolina at Chapel Hill. He is the Chair of the DC Chapter of the Licensing Executives Society, the IP Section of the Bar Association of Montgomery County Maryland, and several organizations, a Fellow of the AIPLA, and teaches IP, technology law, the history of technology and IP. His website is www.rayvandyke.com