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