Thinking about Pi Day has got me thinking about transcendental numbers in general. Nearly all numbers on the number line are transcendental.
Roughly speaking, the transcendental numbers are the ones with no repeating pattern in their decimal expansion. Which means that somewhere in their infinite sequence of digits they contain every possible pattern.
Instead of expressing a number in base 10, we could instead express it in base 96. That would allow us to use every printable character as a digit:
 !”#$%&'()*+,-./0…9:;<=>?@A…Z[\]^_`a…z{|}~
If we do that, then every transcendental number (that is, nearly every number) will contain somewhere in its sequence of digits every literary work that has ever been written or that ever will or could be written.
Which leads to some interesting questions. For example, how many digits do we need to go out in the base 96 representation of Pi before there is a 50% chance that we would come across the complete works of Shakespeare?
I don’t know the answer, but you can be sure it’s a really really big number. I wonder how we would go about calculating that.
Does the character encoding make a difference in the likelihood?
Pi’s expansion in base-96 would also contain all possible math proofs. (Does that means it includes a proof showing pi is in fact rational? Or is that proof impossible and therefore unsequenceable?)
It also contains a sequence that is a map denoting all possible configurations of every particle in every possible (finite and discrete) universe.
It also contains all possible computer programs (in all possible programming languages).
“will contain somewhere in its sequence of digits every literary work that has ever been written or that ever will or could be written”.
Unfortunately, it will also contain every failed rough draft and every discarded idea from those same literary works. A library is valuable not only because of the information it includes, but also because of the worthless noise that it excludes.