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Re: Re: Re: Re: algebraic numbers
*To*: mathgroup at smc.vnet.net
*Subject*: [mg106117] Re: [mg106099] Re: [mg106054] Re: [mg106011] Re: [mg105989] algebraic numbers
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Sat, 2 Jan 2010 05:03:30 -0500 (EST)
*References*: <200912290620.BAA02732@smc.vnet.net> <201001011036.FAA05287@smc.vnet.net>
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
>
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