In honor of pi day, I told my third graders that you could assign a different value to each letter: 01 for A, 02 for B, and so on, up to 26 for Z. Somewhere in pi’s digits is thereby buried the message, “Left Hand of Dorkness, this is a message for you, I’ve hidden it in pi for you, and I am watching you”; this message is repeated elsewhere in pi in every human language from history, and it is repeated for each of my students in every language as well.
But beware, for another message is also found in pi’s digits, repeated in every language: “Don’t get too excited, LHOD, because this is just a byproduct of having an infinite, non-repeating number series.”
I think this is true, but I worry that I’m missing some mathematical principle that renders it false. Am I? Or is it certain that an infinite non-repeating string will have every possible combination somewhere in it?
cool! Next, your third graders should read Borges’ short story “The Library of Babel,” or at least try to find an English sentence in the virtual version.
I don’t think that it has been proven that all possible finite length message are in pi. This is asking if pi is a normal number. People think that it is but it has not been proven.
Pi’s a weird number, and has all sorts of fun twists, but that message actually isn’t true. .12113111411115111116… is another infinite, non-repeating number series that won’t contain your message.
It is not true that any infinite non-repeating string will have every possible combination somewhere within it. You could, for example, have an infinite, non-repeating string in which none of the digits are 3’s.
As for pi in particular, the correct answer is:
Here’s one cite; there are pobably others in those other threads Peter Morris linked to.
I don’t think that any number have been proven to be normal except numbers specifically constructed to be normal like 0.123456789101112131415161718192021222324…
As pi is non-terminating and non-repeating, your assertion isn’t wrong as much as it is absolutely unprovable. Ultimately it’s the million monkeys on a million typewriters with infinite time producing Hamlet question. It’s almost certain to happen at random sooner or later but there is a non-zero possibility that it won’t.
Not a mathematician, but I have a hard time buying the idea that if you have an infinite something X, that it will be bound to contain a certain (something Y) inside it. Saying that pi will have “this is a message for you, I’ve hidden it in pi for you, and I am watching you” as a string in it, seems like saying, “If the universe is infinitely large, then somewhere, there is a planet that is made out of 100% amethyst.”)
In many ways this is exactly the same as the more technical point above. Are there additional constraints on the sequence that limit the sequences that can occur (just as the universe really is unlikely to allow an Amethyst planet)? The thing is that there is no reason to believe that there is. The universe has lots of rules. Pi has essentially none in comparison. Absent constraints, it is reasonable to assume Pi is normal and all possible sequences are allowed. You need to come up with rules that add constraints to justify Pi not being normal. So far, none are known. Which isn’t to say there are none.
Pi does not have “essentially no” rules. It’s 100% rules, and we know all of them. We just don’t understand all of the implications of those rules.
Now, it is known that the vast majority of numbers are normal, and so most mathematicians take it for granted that any number will be normal unless there’s some specific reason for it not to be. We don’t know of any such reason for pi, so it’s usually guessed that it’s normal. And it might in principle be possible to prove it. But we don’t know.
There’s no obvious repeating, and its certainly not rational, so therefore can’t end in repeat the following “digits”…
But its not clear that the patterns aren’t restricted. Eg Only two A’s in 20 letters of the OP’s code… Its possible but the occurrence of the OP’s string is bordering on “no chance” rather than “bound to happen”.
I was alluding to the information content of the rules. Pi has a very small amount of information needed to code what it is. The universe needs a great deal more information to code what its rules are, rules that must include the definition of all the initial conditions and constants to enough accuracy to define the way the universe works - certainly enough to know that amethyst planets won’t (or may) happen.
This is a sort of handwaving Kolmogorov complexity. By essentially no rules I mean we can define Pi very easily. So much easier than the physical universe that the information content of the rules defining Pi are vanishingly small in comparison.
Whenever you want additional structure in something you need more rules - more information in the complexity metric. As Pi has such a low complexity of definition, it is likely that it has very little complexity in its decimal expansion. Which makes it more likely to be normal.
Thanks, all! I learned about normal numbers from this, which is a brand new idea for me. And I think this factoid becomes even weirder if I tell people: it MIGHT be true, but the ratio of a circle’s circumference to its diameter is too mysterious for us to know if it’s true.
While you’re at it … have your students read War and Peace … supposedly there’s a section where every seventeenth word adds up to The Lord’s Prayer … (or was that General Relativity) … either way, it’ll keep them busy 'til the end of the year …