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Rational numbers, rational functions, and Gaussian rationals are examples of fields of fractions. In each of those cases, one knows what the quotients are long before one hears of the idea of constructing the field of fractions of an integral domain.

One case where one (typically??) does not know of such a thing in advance is the field of "convolution quotients"---the field of fractions of a ring of functions of a real variable in which the "multiplication" is convolution.

But convolution quotients will not be appreciated by students who just finished a first-semester calculus course last week. Is there some example one could mention to such students where they wouldn't think they already know what is meant by division of the objects in question?

Later edit suggested by answers and comments posted so far: I had in mind two or three purposes. One was that I wanted to mention this topic a bit obliquely in something the students are to read, and that had to be really terse, so I can't do anything really involved. Less than an hour after I posted the question, this ended up being a parenthetical comment on the course web site that said: "(for example, why is it that one can `divide' one divergent series by another?)". Here I had in mind the ring of formal power series suggested by Chris Eagle, but of course I needed to ruthlessly avoid mentioning power series.

A second purpose concerned possible future uses. Not only in courses: if we get some good examples here, I'd like to add them to Wikipedia's article titled "field of fractions".

A possible third purpose was just the satisfaction of knowing more than one decent example (since the only one mentioned above that's "decent" in the relevant sense is convolution quotients).

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    Maybe formal power series and formal Laurent series?2012-04-26
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    COnstruct the coordinate ring of some curve, define the ring of germs at a point, and then show that you can get it by localizing.2012-04-26
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    @MarianoSuárez-Alvarez : I think your example suffers from the same difficulty that afflicts convolution quotients.2012-04-26
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    +1 Nice pedagogical question. Do the examples need to be *total* quotient fields or can the be proper subrings thereof, such as localizations? It might help to make the title more specific, e.g. replace "certain" by "unanticipated" or "examples... new to novices", or somesuch.2012-04-26
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    At some point thy will have to come to terms with the fact that the notions they are being exposed to are interesting for reasons that thy have not previously been exposed to.2012-04-26
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    @MarianoSuárez-Alvarez : +1 for your last remark. That is actually what I mentioned in the thing I just posted on the course web site. I said (paraphrasing) there are reasons to prefer to avoid fractions when proving a certain statement.... and mentioned that sometimes it's not clear in advance that it's possible to divide the objects in question; hence one can't use fractions.2012-04-26
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    @Mariano: I remember one of my undergraduates teachers said once: "You know you are getting closer to finishing your degree when, instead of hearing a lot of "And this is important for reasons you will see in ..." you hear a lot of "And this is important, for reasons you saw in ...."2012-04-27
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    @ChrisEagle : If you make your comment into an answer with emphasis on the fact that this allows division of _divergent_ series (so that it's not just an example where they _already_ "know" what division is) then I'll "accept" it.2012-05-08

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