Pi Day - The Great Unanswered Questions
Written by Mike James
Thursday, 14 March 2024

It's Pi day again, again, again... Even after so many, I still have things to say about this most intriguing number. The most important things about Pi is that it is irrational and one of the few transcendental numbers we can identify, but you only have to scratch the surface to find questions we don't yet know the answers to.

There are so many interesting properties of Pi that it really does deserve a day to itself - and 3.14 is a very appropriate date,

• Today there are celebrations of Pi going on all over the place, many of them involving eating Pi, but I want to raise some interesting questions about the number we all love to celebrate. These are questions that you might want to try and solve and so make it a real Pi Day. Although I have to say that the chances of you making progress are slim, close to zero I would say and this is a reflection of how little we know about Pi and numbers like Pi.

Recall, lookup if you don't know, that Pi is irrational - that is it cannot be represented by the ratio of two integers. You may think Pi is 22/7, but that's just an approximation. There are far more irrational numbers than rational ones, but proving that a number is irrational is difficult. Pi is also transcendental, which means that there is no finite equation (polynomial) that has Pi as its root. This might seem to be a strange requirement but if a number is the root of an equation this defines the numbers properties. As with irrationality proving that a number is transcendental is difficult.

Proving these thing is hard, but once you know that Pi is irrational the next question is whether simple expressions involving it are rational or irrational. An easy one to answer is "Is pi to a power rational?" The answer is a very obvious NO because if Pin is rational you can use this to construct a polynomial that has Pi as a root and this cannot be as Pi is transcendental.

After an easy answer things fall apart and we are mostly left with open questions. For example, "Is pi+e or pi-e rational?" We still don't know.  Add to this Pie,Pi/e, Pi*e and so on and you can see that we are mostly in the dark about irrational numbers.

A particularly interesting question is "Is there n that makes Pi raised to the power of Pi n times - a so-called tetration - rational?

This old question was raised recently online and has triggered a video that is well worth watching:

You can't get a grip on this one even for n as small as 4 because the value just gets too big. I don't think anyone really expects that computing repeating powers of Pi will end up with a rational number. What is shocking is that we cannot prove it one way or another, nor even have any idea how to start on the problem...

You can find out more in an excellent article on Spektrum.de. It is in German but this is no problem as Google translate does an excelent job. There are also lots of other articles on Pi so today is a good day to visit.

A Pi is mysterious indeed.

• Mike James has more to say on the topic of Pi and transcendental numbers in The Programmer’s Guide To Theorya book which sets out to present the fundamental ideas of computer science in an informal and yet informative way.

Das große Rätsel um die Potenzen von Pi

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