Better, but irrational numbers can have a predictable pattern like "0.110100100010000" ( E.g "0.1 10 100 1000 10000" )
A cryptographically secure pseudorandom number generator lets me pump out a stream of digits that's certainly computable, but unless you know my private key, you won't be able to predict it.
(Finding out the private key from the stream is 'computable', because the definition of computable is comfortable with running exponentially long brute force searches. But that's why 'computable' is not a useful definition in practice. You want something that captures 'tractable', not just 'possible on a Turing machine at all'.)
The most intuitive answer is just computability.
If you have finite memory, you have a finite number of states and will eventually return to a previous state. In your example, you eventually will run out of memory to track the number of consecutive zeros.