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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. But I don't know if this is a "standard" use in numerical analysis.

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?

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    It's not necessarily an "abuse of notions" (or even an abuse of notation); they've just defined a new notation. (Assuming that it is actually defined somewhere explicitly.)2011-09-25

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According to the negative response in MO to this same question, this may be a rather non-rigorous use in mathematics.

As I said in the post, "when one says $O(10^{-6})$, he/she may actually mean $c\times 10^{-6}$ where $1."