Pi Day is about more than pie throwing and pizza deals

Pi is a real number whose most intuitive — and oldest — definition is as the ratio of the circumference of a circle to its diameter. It is also the ratio of the area of a circle to the square of its radius. Its decimal expansion begins 3.14159. 

Approximations to pi were known and used in many ancient civilizations, including Egypt, Babylonia, China, India and Greece. The Greek mathematician Archimedes was the first person known to have shown that the expansion of pi starts with 3.14. He accomplished this by approximating the circumference of a circle by the perimeters of regular polygons inscribed in and circumscribed around the circle.

Pi is ubiquitous in modern science and mathematics. Most people come across it first through the formulas for the circumference and area of the circle or of the volume and surface areas of cylinders and cones. Later, they see it in the study of trigonometric functions. Pi also arises naturally in physical contexts that involve oscillatory behavior, such as springs and pendulums, electromagnetic waves and quantum mechanics.

 Pi has also played an important role in the development of modern mathematics. One of the main concerns of ancient Greek mathematics was geometric constructions via straightedge and compass. In addition to the many constructions that they were able to carry out, such as finding the midpoint of a line segment or bisecting a given angle, there were three problems they were unable to solve: trisecting an angle, duplicating the cube and squaring the circle. Squaring the circle is the problem of constructing a square whose area is the same as the area of a given circle.  In other words, given a circle of radius of 1, can one construct a square with sides of length the square root of pi using a straightedge and compass?

In the 18th and 19th centuries, all three of these classical Greek problems were shown to be impossible, and the methods developed to address these problems were fundamental in the development of modern abstract algebra and geometry.

The most familiar real numbers are the rational numbers — ordinary fractions whose numerator and denominator are integers. These are precisely the real numbers whose decimal expansions either terminate or repeat endlessly.    

Among the irrational numbers, there are some that satisfy some polynomial equation with integer coefficients, while those numbers that do not satisfy such a polynomial are called transcendental.  

Almost all real numbers are transcendental, but it can be very hard to prove that any given number is transcendental. Pi, along with the Euler constant (e=2.71), is the best-known example of a transcendental number. Perhaps we would be celebrating E Day as well if February had at least 71 days.

There are various series expansions for pi that allow one to compute its decimal expansion to an arbitrary number of places, assuming sufficient computing power. The expansion is currently known to 202 trillion digits.

 Competitions are sometimes held on Pi Day to see who can remember the decimal expansion of pi to the most places.

Pi Day gives the opportunity for the mathematics community to explain and celebrate the essential role that mathematics plays in modern life. 

Mathematics research is of fundamental importance to breakthroughs in science and technology. The organization of modern society depends on mathematics, and mathematics provides tools to help people understand the challenges facing the world. However, the contributions of mathematics to the world are often overlooked by the general public, and Pi Day serves as a useful reminder.

 In 2019, building on the tradition of Pi Day celebrations, the United Nations Educational, Scientific and Cultural Organization (UNESCO) proclaimed March 14 the International Day of Mathematics (IDM). The IDM project is led by the International Mathematics Union with the support of many international and regional organizations worldwide.