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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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Slime wrote:
> It's an interesting fact. However, at one point, it occured to me that
> we take radicals for granted as computable.
http://en.wikipedia.org/wiki/Chaitin%27s_constant
How about a number that's trivial to describe in words, has an intuitive
meaning, yet is provably impossible to compute?
--
Darren New, San Diego CA, USA (PST)
Quoth the raven:
Need S'Mores!
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Invisible wrote:
>
> Really? I was sure an algebraic number is any number computable using
> only addition, subtraction, multiplication, division and extraction of
> roots.
http://lmgtfy.com/?q=algebraic+number&l=1
Also see the subsection:
http://en.wikipedia.org/wiki/Algebraic_number#Numbers_defined_by_radicals
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>> Really? I was sure an algebraic number is any number computable using
>> only addition, subtraction, multiplication, division and extraction of
>> roots.
>
> http://en.wikipedia.org/wiki/Algebraic_number#Numbers_defined_by_radicals
Right. I think I see what happened here.
An algebraic number is any number that is the solution to a polynomial
equation. And since I thought that all such numbers are constructable by
radicals, I assumed "solution of a polynomial" and "constructable by
radicals" were equivilent definitions for an algebraic number.
Apparently they aren't equivilent. (Which still seems deeply weird to me...)
As I say, I thought the problem was that no single formula could
encompass the solutions to every possible 5th-degree polynomial. I
didn't realise that there are single, fixed polynomials who's solutions
are simply inexpressible.
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Slime wrote:
>
> 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 suspect that the reason here might be largely historical. IIRC long
ago, the concept of what number could be defined "exactly" was
essentially derived from the Greek notion of which number could be
constructed using only a straightedge and a compass. It turns out that
everything you can write with integers, addition, subtraction,
multiplication, division and square roots can be written in this form.
Thus if you had a number written in this form, you knew that it could be
created with a compass-straightedge construction, but other things like
pi, cube roots, etc. were iffier -- particularly so with roots of
general polynomials. This probably set the stage for looking at
"allowed numbers" as being build by the combination of a set of basic
functions, even when the set of allowed elementary functions was
extended to allow things like nth-roots, exp and log which don't
correspond to circle-straightedge constructions. Nowadays computers
allow us to compute roots to arbitrary polynomials relatively
efficiently, so this way of thinking about things isn't as useful as it
used to be.
Caveat: The above is pure conjecture.
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Invisible wrote:
>>> Really? I was sure an algebraic number is any number computable using
>>> only addition, subtraction, multiplication, division and extraction
>>> of roots.
>>
>> http://en.wikipedia.org/wiki/Algebraic_number#Numbers_defined_by_radicals
>
> Right. I think I see what happened here.
>
> An algebraic number is any number that is the solution to a polynomial
> equation. And since I thought that all such numbers are constructable by
> radicals, I assumed "solution of a polynomial" and "constructable by
> radicals" were equivilent definitions for an algebraic number.
> Apparently they aren't equivilent. (Which still seems deeply weird to
> me...)
You'll get used to it soon enough. I'm sure the result was surprising
to many people when it came out. Beautiful bit of math too.
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> 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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Slime <pov### [at] slimeland com> writes:
> 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
Yes. Once you're in the vicinity of a real root (to keep it simple), you
can calculate it to arbitrary precision.
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>> 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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Kevin Wampler wrote:
> Caveat: The above is pure conjecture.
Heh, all the best ones are. ;-)
I'm still trying to figure out whether there exists a geometric way to
construct logarithms...
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