I was careful. :)
> representable in a finite number of digits or bits
Implying a digit-based representation.
> representable in a finite number of digits or bits
Implying a digit-based representation.
As the other person pointed out, this is representing an irrational number unambiguously in a finite number of bits (8 bits in a byte). I fail to see how your original statement was careful :)
> representable in a finite number of digits or bits
I can define sqrt(5) in a hard-coded table on a maths program using a few bytes, as well as all the rules for manipulating it in order to end up with correct results.
Of course if you know that you want the square root of five a priori then you can store it in zero bits in the representation where everything represents the square root of five. Bits in memory always represent a choice from some fixed set of possibilities and are meaningless on their own. The only thing that’s unrepresentable is a choice from infinitely many possibilities, for obvious reasons, though of course the bounds of the physical universe will get you much sooner.