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,
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 Pi^{n} 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 pie rational?" We still don't know. Add to this Pi^{e},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 socalled 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.
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Last Updated ( Thursday, 14 March 2024 ) 