TITLE: Using decimals of $\pi$ to store data
QUESTION [8 upvotes]: I read recently about an idea to, instead of storing actual data, converting the data to a string of digits and then store the index of where this pattern occurs in some number, for example $\pi$. The idea being that the index of the data would take up less storage space than the actual data.
Of course, we don't know whether $\pi$ is a normal number and hence we do not know if every finite decimal pattern occurs, but let's assume for the moment that it does (or one simply changes to some proven normal number, like the Copeland-Erdős constant).
The thing that struck me was whether the index of the data might actually be a larger number than the data itself. Does there exist some measure of the probability of finding a decimal sequence of length $n$ before the $m$:th decimal place? For $\pi$ in this case, I doubt there's a general formula. Would it depend on the base?
Information or references to other, similar ideas are also very welcome.
(Yes, I understand that this method is very impractical for everyday use, I just found the idea intriguinng.)

REPLY [7 votes]: You don't need to know anything about $\pi$ to answer this one $-$ you just need to know that you can't get something for nothing.
To be more precise: any compression algorithm (which is what your scheme is), if it makes some inputs shorter, must also make some inputs longer. If an algorithm was able to compress, say, all 10000-bit inputs into less than 10000 bits, then the number of possible outputs ($2^{10000} - 1$) would be less than the number of possible inputs ($2^{10000}$) $-$ so you would inevitably have at least one pair of inputs compressing to the same output.