Naked Science Forum
General Science => General Science => Topic started by: thedoc on 23/10/2012 16:25:45
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When will we find out if pi is normal?
Asked by Ravi Nair, Facebook
Visit the webpage for the podcast in which this question is answered. (http://www.thenakedscientists.com/HTML/podcasts/show/20121021/)
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We answered this question on the show...
Matt - That is a very good question because at the moment, we don’t know whether or not pi is normal.
Normal has a very specific definition in maths, so if we say a number is normal, it means that any string of digits is equally likely to appear in its decimal expansion. And so, the decimal places of pi go on forever and they never repeat, but we don’t know effectively how well distributed they are if you go far enough down. And we’ve checked trillions and trillions of digits and it’s been normal all the way so far. But we can't guarantee that it will continue to be normal after that. And so, I suspect it’s going to be a while because currently don’t have any really good mathematical techniques to explore that far into pi or to find some way to pin down to show why it is or it isn’t normal.
But hopefully one day, people will develop some new mathematical ideas and will be able to answer that question.
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"When will ..." questions are mostly impossible to answer. Some can be answered by definition, like "When will the next century start?". Some can be answered with a high degree of confidence, based on well established systems and mechanisms, like "When will the next full moon be?". Answers can be hazarded on a few more -- such questions as "When will it next rain in London?".
The date of a forthcoming (or possibly never coming) mathematical discovery fits none of these categories.
However, I have a scientific hypothesis: a proof that pi is normal will be published on August 23, 2516.
According to an outdated view of the scientific method, I am quite entitled to use that as a working hypothesis until it is disproved. As far as I can see, that could only happen in one of two ways -- arrival of that date, or an earlier proof/disproof of the normality of pi. I do not expect to be around for either event.
[;D]
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So what would be the significance of a number being normal for trillions and trillions of digits, and then suddenly not being normal and showing some kind of pattern or or something, and then maybe going back to normal or not. Has it ever happened before?
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So what would be the significance of a number being normal for trillions and trillions of digits, and then suddenly not being normal and showing some kind of pattern or or something, and then maybe going back to normal or not. Has it ever happened before?
The mathematical concept of normality does not allow for this sort of thing -- it is a holistic concept that involves an infinite set of digits. Not only does each digit have to be present in random (statistically normal) proportions, but each sequence of 4 digits, and each sequence of a trillion digits! I am not a mathematician, but I do not think we have many tools for testing for normality, and I am a little sceptical about whether they will ever become available.
The number that is usually quoted as an example that can be proved to be normal is
0.12345678910111213141516...101102103104...5961596259635964...
which is far from a random sequence of digits, but is nevertheless "normal", and can be shown to be so!