Below is a little math discussion for O(1). It mentioned that using O(1) is not a perfect notation mathematically. Of more interest to mathemeticians than software engineers.
From "Algorithms: Second Edition by Cormen, Leiserson, Rivest & Stein (page 44)"
- Since any constant is a degree-0 polynomial, we can express any constant function as Theta(n^0), or Theta(1). This latter notation is a minor abuse, however, because it is not clear what variable is tending to infinity. We shall often use the notation Theta(1) to mean either a constant or a constant function with respect to some variable. ... We denote by O(g(n))... the set of functions f(n) such that there exist positive constants c and n0 such that 0 <= f(n) <= c*g(n) for all n >= n0. ... Note that f(n) = Theta(g(n)) implies f(n) = O(g(n)), since Theta notation is stronger than O notation.