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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

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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    I have heard people use O in such a way in non-mathematical contexts, but only in words. (I have never seen this *written* as $O(10^{-6})$.) But I assumed that it stems from the english usage ("on the order of hundreds") and not from the mathematical notation. Of course, we cannot draw a precise border between the two.2011-09-21
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    I don't think I've seen any of the books I have use such flagrant abuse of notation. But let me check...2011-09-21
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    usually, for the proportional you write $\propto$ - and mathematically speaking $O(c)$ as a class of functions does not on $c$ if $c>0$.2011-09-21
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    Most of the time I see people using big-O to represent computational complexities, like $O(n)$, $O(n \log n)$, so...2011-09-21
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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