The following question has been on math.SE for several days. Without having a satisfying answer, I'd like to ask the experts here.

In mathematics, the big $O$ notation is used to describe the *limiting behavior* of a function. It is abuse of notation to say
$$
f(x)=O(g(x)).
$$
But this is understandable. However, in the class of numerical analysis, I found that the teacher used the big $O$ notation as the following:

If $\kappa = O(10^{-6})$, $\epsilon_{machine} = O(10^{-16})$, then we can only expect $O(10^{-10})$ accuracy.

I am surprised that they regard $O(10^{-6})$, $O(10^{-16})$, $O(10^{-10})$ as different things. Since according to the definition, they are nothing but $O(1)$.

I guess this is another kind of abuse of notions: when one says $O(10^{-6})$, s/he actually means $c\times 10^{-6}$ where $1<c<10$. But I don't know if this is a "standard" use in numerical analysis and I have never seen this in any paper or articles before.

So here are my questions:

Is there anyone who has seen this kind of usage before? Can anyone come up with the references (books, paper, etc.) that have the similar usage I mentioned above of the big $O$ notation?

[**UPDATE**:]
Before this seemingly stupid question being closed (probably? I don't know), I would like to add my motivation here. I HAVE asked such question in class when the professor wrote down such notation, "what's your underlying limit for the big-O notation, sir?" Unfortunately, I got unsatisfying response: "there is no limit behavior for the notation", and he didn't elaborate any more. This is an applied math course, and it is really frustrating that one cannot understand things rigorously.

Anyway, I can get enough information from the comments in case people don't bother to answer such question.

informally. This is half-way between asymptotic formula (in which one has no information at all about the constant multiple) and a concrete bound (giving complete information about constant factors). Something like $A = O(10^{10})$ can be used to mean that $A$ is either a small factor larger or a small factor smaller than $10^{10}$. The small constant is deliberately left vague. – David Harris Sep 25 '11 at 15:15