The first rarity of Pi is that, that it is not a number. But then, if it’s not a number, what is Pi?

March fourteenth: 3/14 or 3/14 if we read it in English nomenclature. Saying “three fourteen” quickly draws in our mind a number that reminds us of high school. It is also very possible that when listening to it we follow the string: fifteen, ninety-two, sixty-five… as long as our memory reaches us to remember the figures of the singular number Pi.

The US Congress in 2009 officially declared this day, March 14, to be π Day. It was enormously successful from its inception, and the idea grew until in 2019, UNESCO declared it International Mathematics Day . Since then, every year more and more people have joined the celebration, with π as a symbol for those of us who love mathematics.

## Pi is not really a number

Let’s start by clarifying something, Pi is the sixteenth letter of the Greek alphabet (π) and in mathematics we use it to represent something much more interesting than a number (which I am not saying that numbers are not). So, the first rarity of Pi is that, that it is not a number. But then, if it’s not a number, what is Pi?

Pi represents the ratio of the length of the circle to its diameter. A proportion that has the particularity (here its second rarity) of being constant, that is, of always having the same value regardless of how big or how small the circumference is.

In particular, in Euclidean geometry –the one we owe to Euclid (325 – 265 BCE) and which assures us things like that a single line passes through two points– the constant value of Pi is so special (and there are already three) as to be irrational _

It is not that it has lost its reason but, despite being the result of dividing the perimeter by the diameter, it can never be expressed as the division of two integers. If the diameter of a wheel is an “exact” value, without decimals, the space it will cover when making one turn will not be. But then how much will it be? We are approaching a key issue, the value of Pi… but let me first continue with another of its rarities, the fourth now.

Pi is transcendental. It is not that it is so important that it transcends (which also) but that it is transcendent, without n. This mathematical property assures us that Pi will never be the solution of any polynomial. Polynomial? Surely you remember it from your math studies. Polynomials are equations in which the unknown appears raised to one or several natural numbers, for example x ^{2} + x + 3 = 0.

Well, no matter what exponents and numbers are put, there is no polynomial for which x is worth Pi. It is also worth mentioning that this is a property that many numbers do not meet so, at this point, it has already been shown that Pi is rare but the best is still missing. Now yes, let’s talk about its value.

## The elusive value of Pi

As we said at the beginning, the constant value of Pi (in Euclidean geometry) is 3.141592… but, precisely because it is irrational, we know that it will have infinite decimal places. Infinite, as it sounds, without end and, to make matters worse, in this case it is not only that they are infinite but that they do not follow any pattern. They appear to be randomly placed, with all the numbers from 0 to 9 having an equal chance of appearing. In fact, their values can be used as a random number generator and it is possible to search among them for any sequence of figures , even the ID number of any person, who is sure to be found somewhere. However, the most important thing about this property of Pi is that it has become a source of inspiration for the work of many people.

Since the earliest times (there are indications that Pi was already known to the Babylonians in 2000 BC) efforts have been made to establish its value as precisely as possible. In particular, one of t

Archimedes’ idea was followed by many others of a very diverse nature, some even from the point of view of probability and statistics, as was the case of Georges-Louis Leclerc (1707-1788), the Count of Buffon.

In particular, Leclerc found the number Pi while trying to determine how likely it was that throwing a needle over a set of parallel lines would land across one of the straight lines. After various calculations he concluded that if the lines were separated by the same distance as the length of the needle, the probability was 2 divided by Pi. In this way it was easy to approximate Pi by throwing many needles, observing the proportion of these that actually intersected the parallel lines and comparing it with the exact probability.

he first to bear fruit was that of Archimedes of Syracuse (287 – 212 BC), who designed a method to limit the value of this rare constant.

Archimedes used polygons that were inscribed (those that are located inside the circle) and circumscribed (those that contain the circle inside). In this way, the value of the perimeter of the circumference would always be between the perimeter of the inscribed polygon and that of the circumscribed polygon. By adding more and more sides to the polygons, Archimedes managed to give a range of values for Pi, which had a maximum error of 0.040% of the real value… come on, close, close.

However, with the advent of the computer age, the fifth oddity of Pi appeared, being a computable number. In particular, Alan Turing, back in 1936, defined that a number is computable if there is an algorithm that allows us to approximate its value with a number of predetermined decimal places.

## 63 trillion decimal places of Pi have been calculated

Following this premise, in 1949 an ENIAC machine managed to break the record set to date by humans and calculate the first 2037 decimal places of Pi, kicking off a career that has reached 63 trillion (European) figures. with which it was calculated in 2021 by a team from the *University of Applied Sciences* of the Swiss canton of Grisons.

But Pi isn’t just some curious mathematical entity that has been pulling the strings of human thought since ancient times. Pi is, as Rhett Alain assures, an amazing number that appears naturally where we least expect it: in the estimation of our position by GPS, in the movement of the pendulum of a wall clock or even in the way in which an assistant by voice it is capable of recognizing that the user wants, for example, to be told a joke.

But, above all, Pi is the perfect excuse for us to celebrate mathematics and everything it gives us every 14th of March. Happy International Mathematics Day!

*Anabel Forte Deltell , PhD in Mathematics and professor at the University of Valencia, Department of Statistics and Operations Research, University of Valencia*