> I guess the point is, if you say sqrt(2), there are several simple
> algorithms that will allow you to compute exactly what number this is.

Really? Exactly? I would like to see that algorithm! =)


> Whereas if you say "the root of this complicated polynomial", there's no
> immediate way of computing it. (You'd have to say *which* root, for
> starters...)

To my knowledge (which may have big holes in this area), calculating a 
square root and finding the root of a polynomial are both typically done 
with iterative algorithms that converge on a result (and can't give you 
the "true" precise answer). In fact, I think the most common way of 
approximating either one is with Newton's method. You illustrate my 
point: even though calculating a square root and the root of a 
polynomial are similar tasks with similar limitations, you believe that 
sqrt(2) can be "computed exactly" whereas you agree that a root of a 
polynomial cannot.

The problem of *which* root does complicate things some in the 
polynomial case.


> All of these are *representations of* numbers. The question is how
> useful they are, in a given context.

Yeah. We just seem to culturally draw the line between what we consider 
useful and not useful somewhere between radicals and roots of 
polynomials, which aren't very different. I suppose there are reasons to 
have the line there, but I don't think many people realize that it's an 
arbitrary line.

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