# Why is pi special?

2021-03-12

On 14 March every year, the world celebrates the peculiar number pi. Meet the mathematics lecturer Jordi-Lluís Figueras in a conversation about pi and mathematics.

Pi has an infinite number of digits that do not repeat themselves according to any pattern. There are many numbers that have that property, but only pi is designated its own day.

Why is pi so special that it deserves its own day?

“Pi is one of the main constants that appear in mathematics and maybe one of the earliest”, says Jordi-Lluís Figueras, lecturer in mathematics at Uppsala University.

When was pi discovered?

“I suppose, the first time we saw the moon. Every time you see a rounded shape, you see pi. Historically, it was probably the Babylonians, Greeks and Egyptians in between who used pi.”

The first to use an approximation of pi were the Babylonians around 2000 BC. They estimated the pi value to be 3.125 by drawing a hexagon in a circle, calculating the circumference of the hexagon and assuming that the ratio of the circumference of the hexagon to the circle is 24/25. Later on, as the mathematical methods have developed, the approximations of pi have come closer to the real value of pi. But how close to pi can you actually get?

A circle that is 1 cm in diameter has a circumference that is equal to pi. But if we take a measuring tape and measure the circumference, we will get a length that has a finite number of decimals. So not pi. Doesn't this make pi seem a bit unreal?

“Yes and no. Pi is something that lives in the world of mathematics. Now, the question is if what we observe is the same as how we model it. A mathematical triangle is not the same as what we call a triangle in the real world. What we do is that we model the reality by some objects and we take some assumptions that we cannot verify if they are true or not. When we see a rounded object, we say that it’s a circle. And when we say that it’s a circle, we mean that we model this object as a mathematical circle, that is defined by a set of points with the same distance from a given point. But is it true that the real object has the same properties? Can we verify this? I suppose not.

It sounds like it’s impossible to draw a perfect mathematical circle.

“If we take for granted that matter consists of a finite number of particles, your drawn circle will consist of a finite number of particles. So, you’re not drawing a circle, you’re approximating a mathematical circle. In other words, if we suppose that the reality consists of a finite number of particles, then we’ll never be able to measure all the digits of pi.

Thanks to the development of the computer, the number of known digits of pi has increased. According to the Guinness Book of Records, today's record is 50,000,000,000,000,000,000 digits and was set on January 29, 2020 by Timothy Mullican in the United States. The calculation was made using Chudnovsky's algorithm and took eight months. But according to Jordi-Lluís Figueras, the number of known decimals of pi is not so relevant for mathematicians.

“What is important to remember is that pi is an abstract concept that relates very faithfully to physical observations. It enters into a lot of mathematical models that match the reality. By matching, I mean that we experiment and get what the models predict.”

Would it be true to say that the reality is an approximation of mathematics?

“Or the other way around. If I want to model how the population in Uppsala behaves, there are many things that I have to take into account. For instance, Pluto’s gravitation has an marginal effect on us, but in no model you will see Pluto’s orbit as a variable. That’s why you simplify the model and take the most meaningful variables. Otherwise, you are forced to model the whole universe and the model gets so complex that it's out of reach.”

What is your relationship to pi?

“As for any mathematician. Pi appears often and often when we work in mathematics. Sometimes, it shows up naturally, because the computations that you are doing involve some rounded object, like a sphere, so it is expected that pi will be there. And sometimes it appears more mystically. You are computing something where you don’t see any link to a circular object, but pi is there. As a mathematician you reach a certain point where you see that pi always appears. You don’t get super excited when it happens."

How many decimals of pi do you know?

"Eleven. Or twelve. At some point, I decided to learn around ten because I discovered that my students know less. Sometimes, I like to joke with them and ask them how many digits they know. Usually it's four or five. So, to know ten is enough to beat them.

I saw that UNESCO has proclaimed Pi Day to be the International Day of Mathematics, with a different theme each year. This year’s theme is "How can we make the world better with mathematics?". How would you answer that question?

“It’s a big question. There are different aspects of it. One is technological. Take any technological achievement and you’ll se that mathematics is behind it. Another thing that’s a hot topic today, is mathematical modelling of pandemics. Apart from that, there are people who say that if you understand mathematics, you’ll understand logical reasoning, and if you understand logical reasoning, you’ll make better decisions. To that, I am more sceptical. Like anyone else, mathematicians base their support of ideas on emotions.

Are you going to celebrate the Pi Day on Sunday?

“Yes, we always celebrate at home. My wife as a mathematician too.

How will you celebrate?