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Re: Re: Re: algebraic numbers

• To: mathgroup at smc.vnet.net
• Subject: [mg106099] Re: [mg106054] Re: [mg106011] Re: [mg105989] algebraic numbers
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Fri, 1 Jan 2010 05:36:34 -0500 (EST)
• References: <200912290620.BAA02732@smc.vnet.net>
• Reply-to: drmajorbob at yahoo.com

"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

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

• Follow-Ups:
  • Re: algebraic numbers
    • From: Vince Virgilio <blueschi@gmail.com>
  • Re: Re: Re: Re: algebraic numbers
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>