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Re: algebraic numbers
*To*: mathgroup at smc.vnet.net
*Subject*: [mg106154] Re: algebraic numbers
*From*: DrMajorBob <btreat1 at austin.rr.com>
*Date*: Sun, 3 Jan 2010 03:40:58 -0500 (EST)
*References*: <200912290620.BAA02732@smc.vnet.net>
*Reply-to*: drmajorbob at yahoo.com
Mathematica Reals may not be Rational, but computer reals certainly are.
(I shouldn't have capitalized "reals" in the second case.)
Bobby
On Sat, 02 Jan 2010 04:03:30 -0600, Andrzej Kozlowski <akoz at mimuw.edu.pl>
wrote:
>
> On 1 Jan 2010, at 19:36, DrMajorBob wrote:
>
>> "If so, you will indeed have recognized the number x as algebraic, from
>> its first N figures."
>>
>> No... you will have identified an algebraic number that agrees with x,
>> to
>> N figures.
>>
>> OTOH, every computer Real is rational, so they're all algebraic.
>>
>> Bobby
>
> Well, actually Mathematica does not agree with your last assertion:
>
> Head[1.12]
>
> Real
>
> Element[1.12, Rationals]
>
> False
>
> Element[1.12, Reals]
>
> True
>
> In fact it seems clear that the designers of Mathematica have decided to
> interpret all approximate numbers (with head Real) as approximations to
> irrationals rather than as finite expansions of rationals. The only
> rationals in Mathematica are indeed the ones that have the head
> Rational, i.e. fractions.
>
> Andrzej Kozlowski
>
>
>
>>
>> On Thu, 31 Dec 2009 02:17:22 -0600, Robert Coquereaux
>> <robert.coquereaux at gmail.com> wrote:
>>
>>> "Impossible....Not at all"
>>> I think that one should be more precise:
>>> Assume that x algebraic, and suppose you know (only) its first 50
>>> digits. Then consider y = x + Pi/10^100.
>>> Clearly x and y have the same first 50 digits , though y is not
>>> algebraic.
>>> Therefore you cannot recognize y as algebraic from its first 50 digits
>>> !
>>> The quoted comment was in relation with the question first asked by
>>> hautot.
>>> Now, it is clear that, while looking for a solution x of some
>>> equation (or definite integral or...), one can use the answer obtained
>>> by applying RootApproximant (or another function based on similar
>>> algorithms) to numerical approximations of x, and then show that the
>>> suggested algebraic number indeed solves exactly the initial problem.
>>> If so, you will indeed have recognized the number x as algebraic, from
>>> its first N figures.
>>> But this does not seem to be the question first asked by hautot.
>>> Also, if one is able to obtain information, for any N, on the first N
>>> digits of a real number x, this is a different story... and a
>>> different question.
>>>
>>> Le 30 d=E9c. 2009 =E0 18:11, Daniel Lichtblau a =E9crit :
>>>
>>>>
>>>>> To recognize a number x as algebraic, from its N first figures, is
>>>>> impossible.
>>>>
>>>> Not at all. There are polynomial factorization algorithms based on
>>>> this notion (maybe you knew that).
>>>>
>>>> Daniel Lichtblau
>>>> Wolfram Research
>>>
>>>
>>
>>
>> --
>> DrMajorBob at yahoo.com
>>
>
>
--
DrMajorBob at yahoo.com
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