# Another question about the distribution of pi digits (and other famous irrationals).

Fiddling around I came across something I'd like to toss out there. I researched here, and google, but found very little on my specific idea/question. I have the idea, and some empirical evidence, but lack the symbols and math knowledge required to express and prove the idea. Have any of you come across this? What you think overall?

Let p be some decimal place of pi, and let q be the frequency of x e (0,1,2,3,4,5,6,7,8,9) up to p. Then as p -> infinity, q/p -> 1/9 = .1.

It seems to be the case up to 10 million digits of pi. I don't have the resources to test much more. But it also seems to be true for Euler's number, phi, and sqrt2. I know enough about math to be dangerous, I guess. Excuse me if I don't use terms and ideas appropriately as it has been quite some time for me.

Further, consider the set P = {q/p,...} as p -> infinity. Then it seems that P is bounded above and 1/9 is the not only the least upper bound for P, but that the sequence of p/q's converges to 1/9.

That is about all I can say about this. I do have further inquiries however.

1. Is this true for all irrational numbers? I am back and forth on this.
2. Is this important if it is true?
3. Has this been looked or is it established already?
4. What does it mean that pi (and others) adhere to this sort of probabilistic distribution?
5. If it does apply to all irrationals, if we consider the irrationals from say 0 and 1, would there be a uniform shift in how the numbers are distributed?

Ok, I'm spent. Thanks for any replies!

## 3 Answers

1. No. Liouville's constant is irrational yet it only has $0, 1$ in its decimal expansion. And even they are obviously not distributed equally.
2. That depends on how you define "important". Is there anything in this world important?
3. Yes, that has been studied: https://en.wikipedia.org/wiki/Normal_number
4. ? It means that they adhere to this sort of distribution. I don't understand the question.
5. See 1.
• Great points, and thank you. Something to think about. – David Cicero Oct 26 '16 at 18:34
• And yes, things are important!! Cheer up man! You are important! – David Cicero Oct 26 '16 at 19:00
• @DavidCicero I'm not saying they aren't. It's just each one of us has its own view on what is important. :) For example IMO this kind of maths is not important at all. It is extremely fascinating, though. Similar to a beautiful painting. :) – freakish Oct 26 '16 at 19:05

You are looking for the concept of normality. It is conjectured that $\pi$, $e$, and every algebraic irrational (like $\phi$) is normal, even absolutely normal; however, this remains wide open. Indeed, although explicit examples of absolutely normal numbers have been found (Champernowne's constant is normal to base $10$, but not known to be absolutely normal), no naturally occurring real number has yet been shown to be normal.

Meanwhile, it is certainly not true that every irrational is normal. E.g. think about $$0.10110011100011110000...$$ However, it is true that "most" real numbers are normal, in a certain sense - precisely, the set of absolutely normal numbers in $[0, 1]$ has Lebesgue measure $1$. This was proved by Borel in 1909. In another sense, however, most numbers are not normal: the set of non-normal numbers is comeager. This is an interesting example of the ideas of measure and category leading in different directions.

• Nice, thank you. This is good, because I have things to do today and I was completely obsessing. – David Cicero Oct 26 '16 at 18:37
1. This is not true for all irrational numbers. For example: $$0.10110111011110111110\dots$$ is irrational (since it is non-repeating), but has no digits other than $0$ and $1$.

2. This is important, but it is not known if it is true. You have come up with the definition of a (base $10$) simply normal number (though your $1/9$ ought to be $1/10$, since there are $10$ digits from $0$ to $9$). It is not known whether $\pi$ is simply normal (although numerical evidence suggests that it is). It is not even known whether every digit occurs infinitely often in the decimal expansion of $\pi$. These are indeed important research questions - we just don't know the answers yet.

3. See 2.

4. It means that they adhere to this probabilistic distribution - nothing more than that. It is simple to construct normal numbers and it is simple to construct numbers that are not normal. There are not many applications to showing that $\pi$ is normal or not (perhaps to random number generators) - it is an interesting question in its own right.

5. See 1.

• 1/10, yes. I fudged that. – David Cicero Oct 26 '16 at 18:43