>> 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! =)

You can compute it to as much precision as you can compute any other 
result. In practise, that's enough.

>> 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.

To some extent, it's like the difference between a fraction and a 
decimal. The latter is readily comparable and easy to operate on. The 
former... less so. (E.g., is 17/21 > 1247/1918?)

If I say Sqrt(5), it is immediately obvious that one such number exists, 
is strictly positive, and is somewhere between 1 and 5. If I say "x^5 - 
3x^3 + 4x - 6 = 0", it is not immediately obvious how many unique 
solutions exist, whether any of them are purely real (and if so, what 
their sign is), what the magnitudes of these solutions are, or anything 
else. Deducing even these basic facts is quite a bit of work.

In this respect, the Sqrt() function is a kind of "standard form" for 
equation solutions. And by putting things into standard forms of one 
kind or another, they become more useful.

> 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.

Arbitrary, perhaps. (It depends on your definition of "useful", 
obviously.) In this case, it doesn't look especially arbitrary to me.

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