Re: Re: Re: algebraic numbers

*To*: mathgroup at smc.vnet.net*Subject*: [mg106327] Re: [mg106272] Re: [mg106238] Re: algebraic numbers*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Fri, 8 Jan 2010 04:17:28 -0500 (EST)*References*: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net>*Reply-to*: drmajorbob at yahoo.com

> A consequence is that all randoms can be made to fit some algebraic > number of whatever degree, simply by allowing siufficiently large > coefficients. And sufficiently large degree? A combination of the two? Bobby On Thu, 07 Jan 2010 10:19:39 -0600, Daniel Lichtblau <danl at wolfram.com> wrote: > DrMajorBob wrote: >>> Well, I think when you are using Mathematica it is the designers of >>> Mathematica who decide what is rational and what is not. >> Not to repeat myself, but RootApproximant said 100 out of 100 >> randomly chosen machine-precision reals ARE algebraic. >> If your interpretation is correct and consistent with Mathematica, and >> if Mathematica is internally consistent on the topic, virtually all of >> those reals should NOT have been algebraic. >> Mathematica designers wrote RootApproximant, I assume? >> Hence, I'd have to say your interpretation is no better than mine. >> Bobby > > Regarding RootApproximant design, the missing functionality is this. > There is no limiting of coefficient size (or if there is, it's not > obvious to me how it might be done). Rationalize has such limiting > capability, more or less (though it is really built into the algorithm; > the optional second argument does not impose it). > > A consequence is that all randoms can be made to fit some algebraic > number of whatever degree, simply by allowing siufficiently large > coefficients. > > I am not sure whether this is a design flaw. It might alternatively have > been intentional, due to possible implementational difficulties in doing > otherwise. In retrospect, it kinda surprises me that I am not familiar > with the history of this particular design issue, but there you have it. > > Daniel Lichtblau > Wolfram Research > > -- DrMajorBob at yahoo.com