function fields are not the only ones. the most important ones are the p-adic ones. and they have a immensely rich analysis.
So we know that not all fields obey the Archimedean property. For instance, the field of all rational functions with real coefficients does not. But is there any utility in doing analysis in these fields?
Also suppose we have a positive cut and a negative cut . Then we should (and do) get a negative cut. But why do we write it as follows: ?
?? Just having a positive cut and a negative cut doesn't necessarily give us anything! If we have a positive cut, x, and a negative cut, y, then adding the two, which is what you appear to be doing, does not necessarily give a negative cut. But if we have a positive cut, x, and negative cut, y, and |y|> x, then x+ y is a negative cut. As for writing it as -(x)(-y), I suspect that was because they define the sum of two positive cuts first and wanted to write the general case in terms of that.
I am not adding them. I am multiplying them. If and are non-negative cuts, then their product is where and . If we multiply a positive cut by a negative cut, we get a negative cut. But why is this represented as ? Because this "looks" positive (e.g. a negative multiplied by a negative is a positive).