What Is the Significance of Pi in Mathematics?

I’m Andrew Barnes, an applied mathematician at the GE Global Research Center. As part of our celebration of Pi day on March 14, I’m going to talk a bit about Pi. I work on a variety of stochastic and probabilistic problems at GE, and in all these problems the number Pi is fundamental and ubiquitous. In fact, Pi is just as important to me in my work as the English alphabet is to anyone who reads and writes in English. There is not a day that passes where I do not use Pi  in some shape or form, or in some calculation.

The reason for the deep and widespread use of Pi in my work (or the work of any mathematician) is that Pi is a fundamental mathematical building block. In fact it is so fundamental that it pervades human existence as intelligent beings; the number Pi plays an important role in many of the major scientific advances of mankind. In this post, I will present a few mathematical vignettes of Pi, showing how Pi marks major milestones of human intellectual history from antiquity to the present.

1. What is Pi? Existence questions and Euclid
Early on in school, we learn that Pi is the ratio of the circumference of a circle to its diameter. The interesting thing about this is that this ratio is the same for all circles, whether it is as small as a pea or as big as the sun. Although this fact was probably recognized by all the ancient civilizations, its rigorous statement and proof was probably first written by Euclid in his seminal work Elements. So Euclid marks the stage in human civilization by which we were able to prove the existence of Pi.

2. Gregory-Leibniz Series
We now jump several centuries to the period of Issac Newton and his contemporaries. Although various approximations for Pi had been in use for several centuries, this is a period where infinite series were used to provide an exact representation of Pi, as well as derive accurate approximations. One of the most celebrated of these infinite series representations is the following:

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This series is called the Gregory-Leibniz series (discovered separately by Gregory in 1671 and Leibniz in 1674), and was also used by Isaac Newton to approximate Pi to 15 digits. This series depends on the Taylor expansion of the function f(x)= arctan (x), and this requires the development of calculus — which is another significant landmark in human intellectual history (for which Newton and Leibniz share credit).

3. Basel Problem and Leonhard Euler
A celebrated question that remained unsolved for almost a century is the Basel problem. It consists of finding the sum of squares of reciprocals of natural numbers (i.e. the right hand side of the equation below):

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The interesting point here is that the answer is one-sixth of the square of Pi. This was first discovered by Leonhard Euler in 1735 (and proved rigorously also by Euler in 1741). This surprising connection between the Basel problem and Pi has a very clean modern explanation using Fourier analysis, another major intellectual landmark based on Joseph Fourier’s work on heat conduction in 1822.

4. Squaring the circle, Transcendence of Pi
A classical problem dating to antiquity is the question of squaring the circle. This problem consists of constructing a square equal in area to a given circle using only a straightedge and compass and a finite number of steps. This turned out to be a rather difficult problem that was unsolved until 1882, when Lindemann proved that squaring the circle was impossible. The proof consists of showing that Pi is transcendental, which means that there is no polynomial with rational coefficients whose solution is Pi.

The Lindemann proof and its subsequent generalizations bring us to topics of a very modern flavor in mathematics. For example, the analog of the Lindemann theorem with Pi-adic numbers instead of the usual algebraic numbers is a currently unsolved conjecture.

5. Digital computers and the quest for digits of Pi
The advent of digital computers in the mid twentieth century led to the development of techniques to compute very accurate approximations of Pi. The purpose of these computations was not really for practical applications — it is hard to think of any physical or engineering application that needs more than (say) a hundred digit approximation of Pi for any reason. So the pursuit of computing increasingly accurate rational approximations of Pi is an intellectual exercise for its own sake. John von Neumann used the ENIAC (the first digital computer) to compute over 2000 digits of Pi and subsequent decades of mathematicians and computer scientists have extended this to our current capability of 10 trillion digits.

6. Ramanujan and his goddess
The ingredients for the multi-million digit calculations of Pi  are a combination of iterative algorithms and some truly ingenious infinite series. Srinivasa Ramanujan (1887 — 1920) was a self-taught mathematical prodigy who came up with several such series, such as:

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Ramanujan produced several diaries of formulae of this kind, in which he simply stated formulae with no proofs or even hints of their justification. These diaries have kept generations of subsequent mathematicians busy in a quest to prove the unproven formulae. They are remarkable for their mathematical depth, insight and beauty. Efforts to prove some of these formulae have led to the development of entirely new areas of mathematics. Ramanujan’s insight is indeed hard to explain – his own explanation of the source of his formulae is that his family goddess Namagiri revealed them to him!

Now that you’ve learned a bit about the history of Pi and why it is so important to my daily life as a mathematician, check out the below video, the first of the sPI CAM series capturing two GE statisticians using Pi to help solve a problem for GE Capital. Later in the day, we’ll be sharing more footage of Pi being used at the research center, captured by the sPI CAM!

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