Why is it O(1) (and not, say, O(2))? If the running time of an algorithm scales linearly with the size of its input, we say it has $O(N)$ complexity, where we understand N to represent input size.

If the running time does not vary with input size, we say it's $O(1)$, which is essentially saying it varies proportionally to 1; i.e., doesn't vary at all (because 1 is constant).

Of course, 1 is not the only constant. Any number could have been used there, right? (Incidentally, I think this is related to the common mistake many CS students make, thinking "$O(2N)$" is any different from $O(N)$.)

It seems to me that 1 was a sensible choice. Still, I'm curious if there is more to the etymology there—why not $O(0)$, for example, or $O(C)$ where $C$ stands for "constant"? Is there a story there, or was it just an arbitrary choice that has never really been questioned?
"$O(1)$" is used because it is simple, clear and unambiguous. "$O(C)$"&nbsp;would be a poor choice of notation because in any given context, $C$&nbsp;might have a specific meaning, such as the number of clauses in a CNF formula, the number of components in a graph.