I’ve been pondering an interesting question for a while and can’t come up with a satisfactory answer.
Consider the number pi. Also consider the works of Shakespeare, converted into a string of numbers where each number represents a letter (a=1, b=2, or you could use ASCII codes, or whatever). In fact, we can consider any series of numbers that’s well-defined, but Shakespeare makes it more fun.
The question: is it possible that Shakespeare is encoded within pi somewhere?
In a larger sense, the question is whether an arbitrary sequence of digits can be found within pi. You can even use different bases, if that suits your fancy. The problem doesn’t really change.
At first, I thought the answer was definitely “yes”. After all, pi is infinitely long and has a (seemingly) random ordering of digits. Somewhere in that mess, perhaps septillions of digits down the line, surely you can find Shakespeare.
My hunch wasn’t strong enough proof, so I began to consider how to disprove it. In other words, can I construct an infinitely long irrational number and a sequence of numbers that does not appear anywhere within it? At first I considered a non-repeating sequence of, say, ones and zeros that increases its count on each iteration. To wit:
1.010011000111000011110000011111…
It’s clear that the sequence “123” will never appear in this number. I believe the number is irrational, because it cannot be represented by a fraction– or, more loosely, because it’s infinite and never repeats. Thus, Shakespeare will never be found in here.
But pi is a slightly different animal: it has a fairly even distribution of digits– if I remember right, there is almost exactly a 10% occurrence of each digit zero through nine in the first million digits of pi. However, I don’t believe it’s been proven that the distribution is even (meaning pi is a “normal” number), and thus it’s entirely possible that somewhere down the line it becomes all sixes and sevens or something.
Anyway, I want to believe Shakespeare is in there, but I can’t come up with a strong conclusion.
(Yes, this sort of thing keeps me awake at night.)