First we must decide on which “Hamlet” we are talking about. There were quite a few different texts of Shakespeare’s plays, especially in the early days, and the number of words in each were not always exactly the same.
Also, keep in mind that Shakespeare never used the letter j.
How’s that for hijacking my own thread?
Three.
”Words, words, words.”
Ham, Let, Tale, Male, Math, Lathe, Mate, Hate, Late, Metal, Meal, Meat, …
Wouldn’t you know it? I have a Shakespeare concordance that just a few days ago I put in a pile to give away because I’ve never used it. And now a need for it arises.
According to my concordance, “words” occurs 20 times in Hamlet, and “word” occurs 12 times.
Hat, Lame, Am, Eat, A, Hale, Me, He, Hat, Malt, Mat, Ale, The, Them, Met, Ate, …
Tam, Lam, Hem, At, Heal, Heat, Elm, Tame, Halt, Lath, …
Hamlet, Hamlet, bo-bamlet, me-mi-mo Mamlet. Hamlet!
Yes, you can make probabilistic claims about fixed events, for example without knowing anything about it I say that Trump’s blood type has about 45% probability of being type O, even though in reality that it either is or isn’t. But this requires certain assumptions like Trump is just as likely to be of type O as any other Caucasian living in the US.
We could do similarly with pi and say that with respect to the distribution of its digits its probably no different from any other transcendental number, and since those are almost surely normal it is too. This is why mathematicians believe that pi is normal. But that assumption is not proved, and pi isn’t a random number it is a fixed number with certain unique qualities, which very well could include not being normal.
At this point, that’s probably not a problem.
That’s all I am saying. The post I replied to seemed to be saying that one can’t talk about the digits of pi probabilistically because those digits are “determined”. I was just noting that in some common interpretations of probability, this isn’t a problem at all.
Normality isn’t relevant to that.
Tea, Eth, Teal, Melt, Team, …
I was about to say that if pi were normal then there would be no limit on the number of zeros in a row but now I don’t think that is the case. Suppose there is a limit on the number of zeros in a row in pi when expressed in base 10. So long as the same limit applies to all of the other digit values, pi would still be normal. Right? Maybe it depends on your definition of normal. I just used Wikipedia’s definition because it is almost bedtime.
The standard definition of normal is that, given any length of string, all strings of that length are asymptotically equally likely. There certainly exist strings of length a million in pi, and one possible string of length a million is a million zeroes, and there are a finite number of such strings, so there must be strings of a million zeroes. Likewise for length a billion, or length a googol, or length TREE(3).
EDIT: I think you might have mistaken “simply normal” for “normal”. “Simply normal” only applies to counts of single digits, not to strings of digits.
A question comes to mind that a cursory scan doesn’t reveal the answer to: There are numbers that are known to be normal in some base, and numbers that are known to not be normal in some base, and there are known to be numbers that are normal in every base, and there are even some specific (but uncomputable) numbers that are known to be normal in every base. But: Is there known to be any number that is normal in any one base, but not normal in any other one base? That is, is the set of numbers normal in any given base known to be distinct from the set of absolutely-normal numbers?
If you look at the beginning of the proceedings of the workshop on Computational and Analytical Mathematics in May 2011, they mention
\displaystyle
\sum_{k=1}^\infty \frac{1}{3^k 2^{3^k} }
as an explicit constant that is normal in base 2 but not base 6.
If there exist positive integers r and s such that p^r = q^s , then any p-normal number is also q-normal, but if not then there exist uncountably many numbers that are p-normal but not q-normal.
I concede to the distinction between “simply normal” and "normal” However, if I had said, ” Suppose there is a limit on the number of zeros in a row in pi when expressed in base 10. So long as the same limit applies to all of the other digit values, pi would still be simply normal.” then that might be correct?
Helm, Meta, Meh, Em, …
Has anyone ever tried to prove that it is impossible to determine whether or not pi is normal?
Hmm, nice wording. The “impossible to determine” aspect does occur for some problems in Math (and Computer Science). Among the the most famous are Gödel’s incompleteness theorems where it is proven that certain things cannot be proven.
The examples of such proofs that I am aware of cover fairly general problems where one can “world build” complicated equations or computational models. The “Is Pi normal question?” doesn’t appear to be general enough to allow such world building.
I consider 2^83 as “infinity for all practical purposes” when counting something. E.g., digits of Pi. That’s about 10^24. If the digits of Pi are reasonably distributed (they cannot, of course, be random) any sequence of length 24+ that’s independent of the digits of Pi that you are looking for in Pi will never be found.
(I wonder how long this thread can go before someone mentions Liouvillle numbers. Wait. Oh, drat.)