The constant depends on the two languages, but not on the number. As army1987 points out, if you pick the number first, and then make up languages, then the difference can be arbitrarily large. (You could go in the other direction as well: if your language specifies that no number less than 3^^^3 can be entered as a constant, then it would probably take approximately log(3^^^3) bits to specify even small numbers like 1 or 2.)
But if you pick the languages first, then you can compute a constant based on the languages, such that for all numbers, the optimal description lengths in the two languages differ by at most a constant.
The context this in which this comes up here generally requires something like “there’s a way to compare the complexity of numbers which always produces the same results independent of language, except in a finite set of cases. Since that set is finite and my argument doesn’t depend on any specific number, I can always base my argument on a case that’s not in that set.”
If that’s how you’re using it, then you don’t get to pick the languages first.
You do get to pick the languages first because there is a large but finite (say no more than 10^6) set of reasonable languages-modulo-trivial-details that could form the basis for such a measurement.
The constant depends on the two languages, but not on the number. As army1987 points out, if you pick the number first, and then make up languages, then the difference can be arbitrarily large. (You could go in the other direction as well: if your language specifies that no number less than 3^^^3 can be entered as a constant, then it would probably take approximately log(3^^^3) bits to specify even small numbers like 1 or 2.)
But if you pick the languages first, then you can compute a constant based on the languages, such that for all numbers, the optimal description lengths in the two languages differ by at most a constant.
The context this in which this comes up here generally requires something like “there’s a way to compare the complexity of numbers which always produces the same results independent of language, except in a finite set of cases. Since that set is finite and my argument doesn’t depend on any specific number, I can always base my argument on a case that’s not in that set.”
If that’s how you’re using it, then you don’t get to pick the languages first.
You do get to pick the languages first because there is a large but finite (say no more than 10^6) set of reasonable languages-modulo-trivial-details that could form the basis for such a measurement.