The Rarities of Pi, the Number with the Most Fans in the World

March Fourteenth: March 14 or March 14 if read in English nomenclature. When we say “three fourteen,” we quickly recall a number that reminds us of high school. It is also very possible that while listening to it, we follow the chain: fifteen, ninety-two, sixty-five… until our memory reaches us to remember the digits of the singular pi.

The US Congress in 2009 officially declared this day, March 14, π Day. It was a huge success from the start and the idea evolved until UNESCO declared it International Mathematics Day in 2019. Since then, more and more people have joined the celebration every year, and the number pi has become a symbol for those of us who love math.

Pi is not really a number

Let’s start with a clarification, Pi is the sixteenth letter of the Greek alphabet (π), and in mathematics we use it to mean something much more interesting than numbers (I’m not saying numbers aren’t numbers). So the first rarity of pi is that it’s not a number. But then, if it’s not a number, what is pi?

Pi is the ratio of a circle’s length to its diameter. A proportion whose peculiarity (here its second rarity) is that it is constant, that is, it always has the same value, no matter how large or small the circle is.

Particularly in Euclidean geometry—the one we owe to Euclid (325–265 B.C.) and which assures us that one line passes through two points—the constant value of pi is so special (and their three) what to be irrational

It’s not that it has lost its meaning, but despite the fact that it is the result of dividing the perimeter by the diameter, it can never be expressed as a division of two integers. If the wheel diameter is an “exact” value, without decimal places, then there will be no space that it will cover in one revolution. But then how much will it be? We come to the key question, the value of pi… but let me first move on to another rarity, now the fourth.

Pi is transcendent. What is important is not that it is transcendent (which is also), but that it is transcendent, without n. This mathematical property guarantees us that pi will never be the solution of any polynomial. Polynomial? You probably remember this from your math classes. Polynomials are equations in which the unknown appears as one or more natural numbers, such as x.² + x + 3 = 0.

Well, no matter what indicators and numbers are set, there is no polynomial for which x is worth pi. It’s also worth mentioning that this is a property that many numbers don’t match, so Pi has already been shown to be rare at this point, but the best is still missing. Now yes, let’s talk about its value.

The Elusive Significance of Pi

As we said at the beginning, Pi’s constant value (in Euclidean geometry) is 3.141592… but precisely because it’s irrational, we know it will have infinite decimal places. Endless, as it sounds, without end, and what’s worse, in this case, not only are they endless, but they don’t follow any pattern. They seem to be randomly placed and all numbers from 0 to 9 have an equal chance of appearing. In fact, their values ​​can be used as a random number generator and among them you can search for any sequence of numbers, even the identification 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 ancient times (there are indications that the number Pi was known to the Babylonians already in 2000 BC), attempts were made to establish its value as accurately as possible. In particular, one of the first to bear fruit was Archimedes of Syracuse (287–212 BC), who developed a method for limiting the value of this rare constant.

Archimedes used inscribed (those inside a circle) and inscribed (those containing a circle inside) polygons. Thus, the value of the perimeter of a circle will always be between the perimeter of the inscribed polygon and the perimeter of the circumscribed polygon. By adding more and more sides to the polygons, Archimedes was able to obtain a range of pi values ​​that had a maximum error of 0.040% of the real value… come on, close it, close it.

The idea of ​​Archimedes was followed by many other ideas of a very different nature, some even in terms of probability and statistics, such as Georges-Louis Leclerc (1707-1788), Comte Buffon.

In particular, Leclerc found pi by trying to determine the probability that, by throwing a needle through a set of parallel lines, it would land on one of the lines. After various calculations, he came to the conclusion that if the lines are separated by a distance equal to the length of the needle, then the indicated probability is 2 divided by pi. Thus, one could easily approximate Pi by throwing many needles, observing the fraction of them that actually crossed the parallel lines, and comparing that to the exact probability.

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

63 trillion decimal places of pi calculated.

Following this premise, in 1949 the ENIAC machine succeeded in breaking the record held by humans today and calculating the first 2037 decimal places of pi, launching a career that reached 63 trillion (European) digits with which it was calculated in 2021 . a team from University of Applied Sciences Swiss canton of Grisons.

But Pi is not just a curious mathematical unit that has pulled the strings of human thought since ancient times. Pi is, as Rhett Allen assures us, an amazing number that naturally shows up where we least expect it: in GPS estimating our position, in the movement of a wall clock pendulum, or even in how the voice assistant is able to recognize , for example, that the user wants to be told a joke.

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

Anabel Forte Deltel, Doctor of Mathematics and Professor, University of Valencia, Department of Statistics and Operations Research, University of Valencia

This article was originally published on The Conversation. Read the original.

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