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>