[tzs]

That kind of reminds me of the idea I've heard several times of compression by indexing π. The idea is that every possible finite byte sequence occurs somewhere in π [1] so if you've got some file of N bytes that you want to store you just need to store N and an index to a place in π where that sequence of bytes occurs.

The problem is the the index to the first occurrence of any particular N byte sequence will on average take around N bytes to store so your "compressed" file is about the same size as your input file.

[1] This is not known to be true. A number that has the property that every possible finite sequence of digits in base b occurs in the number's base b expansion is called a "normal number in base b". A number that is normal in base b for all integer bases b ≥ 2 is simply called a "normal number". Almost all real numbers are normal, but it is not known if π is among them.

[Dylan16807]

> Almost all real numbers are normal

Well if we're being picky then I'm going to point out that almost all real numbers cannot be written down.

Of the numbers we can write down in a reasonable way, the computable numbers, we've only proven a relative handful of them to be normal in any base, and barely any normal in all bases.

So if you have an arbitrary irrational number where you can ask if it's normal, then the answer is probably "we don't know, we haven't figured out a proof", rather than "almost certainly". Those overwhelming odds about "almost all real numbers" don't apply to your number, because your number is computable.