Slime wrote:
>  > Hmm, interesting. I thought the theorum states that there is no single
>  > formula that covers all possible polynomials of a given degree. However,
>  > Wikipedia asserts something far stronger: It seems to claim that you can
>  > construct a single, fixed polynomial who's solutions (which are also
>  > fixed) cannot be expressed by radicals.
> 
> It's an interesting fact. However, at one point, it occured to me that 
> we take radicals for granted as computable. Consider sqrt(2). Somehow, 
> we're OK with expressing a number as "the number which, when squared, 
> equals two." Yet, we have no way of expressing it otherwise. We've all 
> seen approximations to this number, we would recognize it if we saw the 
> first few digits, yet we can't actually communicate it without 
> describing it in terms of the solution to x*x = 2. (There may be other 
> properties it has that we could use to describe it, but we still can't 
> just write it down and point to it, like we can with, say, an integer. 
> We can never find its precise value, we can only approximate it.)
> 
> So, it's kind of interesting that when we can't come up with a way to 
> describe a number besides "one of the roots of this degree-5 
> polynomial," we typically consider that less descriptive than the value 
> of an expression with radicals.

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. 
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...)

The root notation is kind of a standardised way of mentioning the roots 
of particularly simple polynomials.

(Indeed, if you say Solve[x^5 - x + 1, x] to Mathematica, it replies 
with something like x = Root[1 - #1 + #1^5, 1], which simply means "the 
first root of x^5 - x + 1".)

Then again, A/B means "the solution to Bx - A = 0". And if you want to 
be pedantic about it, 768 means 7*10^2 + 6*10^1 + 8*10^0.

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

Standard roots are useful in that, for example, Sqrt(8) is clearly 
larger than Sqrt(2). But given Root[1 - #1 + #1^5, 1] and Root[1 - #1 + 
#1^5, 2], which one is larger? (Actually, at least one of those is 
complex, so it's probably not a meaningful question.) Decimal 
approximations are less accurate than fractions, yet vastly easier to 
compare and do arithmetic with. And so on.

Usually, the roots of an arbitrary polynomial aren't very easy to work 
with. But then, a 20-mile formula is no less easy to work on either.

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