Coming from a post on theydidthemath talking about Pi, all the top answers begin by explaining that **if** we consider Pi to be a “normal number”, then etc. They also mention we haven’t been able to prove whether it’s a normal number or not yet. What’s that about?
In: Mathematics
What’s really interesting about normality, is that it is known that almost all numbers are normal. Which means that if you throw a dart at random on the interval [0,1] , it would fall, with probability 1, on a normal number.
Not being normal is rare, because to have a non-normal number, you need to have some kind of pattern in its expansion in some base, and patterns do not occur when you pick a number at random.
We believe that most irrational constant that we know (pi, e, sqrt(2), cos(1), solutions of interger polynomial equation, etc …) are normal, because we see no reason to have a pattern in their expansion and computer experiments suggest that there are none. But so far, we never could prove that any of these number are really normal, even in some specific base.
The only numbers that we know to be normal in some base are constructed specifically in the purpose of having that property. For example the Champernowne constant 0,123456789101112131415161718192021222324…. is normal in base 10. But we don’t even know if it’s normal in any other base …
But what’s even more striking is that we have absolutely no explicit example of a number that would be normal (in any base), even if we know that most of them are. The only “implicit” examples that I know of specific normal numbers are [Chaitin constant](https://en.wikipedia.org/wiki/Chaitin%27s_constant) which are not computable …
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