Re: Re: Re: Re: algebraic numbers

*To*: mathgroup at smc.vnet.net*Subject*: [mg106280] Re: [mg106238] Re: [mg106220] Re: [mg106192] Re: algebraic numbers*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Thu, 7 Jan 2010 02:30:28 -0500 (EST)*References*: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net>*Reply-to*: drmajorbob at yahoo.com

If I'm told that finite-precision reals are not Rational "because Mathematica says so", but that Mathematica success (by some algorithm) in finding a Root[...] representation doesn't mean the number is algebraic... yet I know that all finite binary expansions ARE both rational and algebraic as a matter of basic arithmetic... then I question whether Mathematica is saying anything either way. Perhaps it's just Mathematica USERS holding forth in each direction. I think the view of reals as monads (a la nonstandard analysis) melds with the fact that reals are irrational A.E. and non-algebraic A.E., while monads are, of course, consistent with the spirit of Mathematica's arbitrary-precision arithmetic (WHEN IT IS USED). The OP posted a number far beyond machine precision, so it's reasonable to come at this from that arbitrary-precision world-view... in which case you're "right" and I'm "wrong". I called all the reals rational, and you called them monads (or equivalent). Fine. Bobby On Wed, 06 Jan 2010 16:46:20 -0600, Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote: > > On 7 Jan 2010, at 04:19, 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. > > No, they are not real algebraic. RootApproximant gives algenraic > approximations to these numbers and in fact it uses a test for what > makes a good approximation. In never says that these numbers themselves > are algebraic. You have been completely confused about this. The method > RootApproximant uses is the LLL method, which finds approximations. > Because of this it will give you a number of different approximations > for the same real. For example > > In[7]:= RootApproximant[N[Pi, 10], 2] > > Out[7]= (1/490)*(71 + Sqrt[2156141]) > > In[8]:= RootApproximant[N[Pi, 10], 3] > > Out[8]= Root[37 #1^3-114 #1^2-36 #1+91&,3] > > So how come N[Pi,10] is equal to two quite different algebraic numbers? > You should first understand what an algorithm does (e.g. > RootApproximant) before making weird claims about it. (In fact Daniel > Lichtblau already explained this but you just seem to have ignored it). > > Andrzej Kozlowski > >> >> 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 >> >> On Wed, 06 Jan 2010 04:57:26 -0600, Andrzej Kozlowski >> <akoz at mimuw.edu.pl> wrote: >> >>> Well, I think when you are using Mathematica it is the designers of >>> Mathematica who decide what is rational and what is not. >>> >>> And when you are not using Mathematica (or other similar software which >>> interprets certain computer data as numbers), than I can't imagine what >>> you could possibly mean by a "computer number". >>> >>> Andrzej >>> >>> >>> On 6 Jan 2010, at 11:45, DrMajorBob wrote: >>> >>>> Obviously, it DOES make them rational "in a sense"... the sense in >>> which I mean it, for example. >>>> >>>> Bobby >>>> >>>> On Tue, 05 Jan 2010 20:41:34 -0600, Andrzej Kozlowski >>> <akoz at mimuw.edu.pl> wrote: >>>> >>>>> >>>>> On 6 Jan 2010, at 11:13, DrMajorBob wrote: >>>>> >>>>>> I completely understand that Mathematica considers 1.2 Real, not >>> Rational... but that's a software design decision, not an objective >>> fact. >>>>> >>>>> I think we are talking cross purposes. You seem to believe (correct >>> me if I am wrong) that numbers somehow "exist". Well, I have never seen >>> one - and that applies equally to irrational and rationals and even >>> (contrary to Kronecker) integers. I do not know what the number 3 looks >>> like, nor what 1/3 looks like (I know how we denote them, but that's >>> not >>> the sam thing). So I do not think that the notion of "computer numbers" >>> makes any sense and hence to say that all computer numbers are rational >>> also does not make sense. There are only certain things that we >>> interpret as numbers and when we interpret them as rationals they are >>> rationals and when we interpret them as non-computable reals than they >>> are just that. >>>>> Of course we know that a computer can only store a finite number of >>> such objects at a given time, but that fact in no sense makes them >>> "rational". >>>>> >>>>> Andrzej Kozlowski >>>> >>>> >>>> -- >>>> DrMajorBob at yahoo.com >>> >>> >> >> >> -- >> DrMajorBob at yahoo.com > -- DrMajorBob at yahoo.com

**Re: algebraic numbers**

**Re: Re: Re: Re: algebraic numbers**

**Re: Re: Re: Re: algebraic numbers**

**Re: Re: Re: Re: algebraic numbers**