Re: Re: algebraic numbers

*To*: mathgroup at smc.vnet.net*Subject*: [mg106323] Re: [mg106295] Re: algebraic numbers*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Fri, 8 Jan 2010 04:16:42 -0500 (EST)*References*: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net>*Reply-to*: drmajorbob at yahoo.com

> And in > particular, 1.2==12/10 in Mathematica should trouble you if you believe > Mathematica speaks meaningfully on these issues. That's it, in a nutshell. Bobby On Thu, 07 Jan 2010 01:33:33 -0600, Richard Fateman <fateman at cs.berkeley.edu> wrote: > Andrzej Kozlowski wrote: >> Just one more comment, I hope my last one on this subject. Obviously >> RandomReal make it choices out of a countable set of entities. One would >> have to be insane to claim otherwise and I am not that yet. >> >> But, Mathematica does not regard these entities as rational numbers and >> so they are not that. If you call them rationals the it does not make >> *mathematical* sense (because rationals have measure 0). So, if >> Mathemaitca does not regard them as rationals they are not rationals. > > You can refer to Rationals as whatever Mathematica > calls Rationals. But the rational numbers include all numbers that are > represented by finite explicit binary strings in a floating-point > format. They also include other numbers whose binary expansions are > infinite, but repeat. > > Can Mathematica represent Reals that are NOT RATIONAL? Sure. Here are > examples: Sqrt[2], 3*Pi, 4*E. 3*E +4*E^E + 5*E^E^E. > Incidentally, it is not known if E+Pi is rational. > >> How could they be that ? Until they are interpreted by Mathematica, they >> are not numbers at all but just some data stored in computer memory - >> which are not numbers of any kind. > true, but other programs can also interpret them. As numbers, as ASCII > character strings, as pointers into memory. > Mathematica interprets them as >> non-computable irrationals > > No, that's not the way computer programs work. Mathematica allows some > set of operations like +, *, printing. That's all. They are obviously > computable and finitely representable as well, but Mathematica doesn't > need to have an opinion on this, and neither do we have to attribute > opinions to Mathematica. > > If you think that the operations that Mathematica performs are > consistent with YOUR view that these numbers are non-computable > irrationals, I suppose that is your view, but it is certainly > unnecessary for others to hold this view. > > in order to make mathematical sense when >> returning them while simulation a real distribution, because all other >> numbers have measure 0. > > There is a literature on pseudo-random numbers that makes mathematical > sense without any such interpretation. >> >> This is all about "simulating mathematics" - numbers do not live in any >> sense inside computers. To say that "all computer numbers are rational" >> is weird - there is no such things as "computer numbers". Numbers exist >> only and (probably) exclusively in the human mind. > > Actually, you just said that Mathematica interprets --blah blah. Maybe > you think that Mathematica has a human mind? >> >> To say that 1.2 is rational in Mathematica even if Mathematica says >> >> Element[1.2, Rationals] >> >> False >> >> does not make any sense at all. > > It makes perfect sense to say (in Mathematica) that 1.2 is a rational > number because it is equal to a rational number. Huh?? > 1.2==12/10 > True. > > (A better example would be 1.25, since 1.2 is not representable exactly > in binary. This example of 1.2 actually reveals a "misfeature of > mathematica. > > 1.2==5404319552844595/4503599627370496 > True. > > So 1.2 is actually Mathematica-equal to another rational number. Many, > in fact. > ) > > > If you capitalize the term and wish to say that 1.2 is not a Rational > in Mathematica, that is just a convention based on the "type" of data > that is input to Mathematica with a decimal point and is therefore > stored in a memory format that is labeled "Real" which (in Mathematica) > is a superclass of "Rational". That is, "1/2" is a Rational but is also > a Real and is incidentally also a Complex. But 0.5 is not a Real. > > This categorization of types in Mathematica does not determine the > membership (or not) of a particular numeric VALUE in a mathematical > category such as "rational". From a mathematical perspective, any > legal number in an ordinary floating-point format can represent only a > rational value. > >> >> Andrzej Kozlowski >> >> >> >> >> On 6 Jan 2010, at 07:04, Andrzej Kozlowski wrote: >> >>> Well, you are obviously misunderstanding what I am trying to explain >> but I have no desire to spend any more time on it. I give up. > > OK. > >>> Perhaps you should try to explain yourself why Mathematica gives >>> >>> In[1]:= Element[1.2, Rationals] >>> >>> Out[1]= False >>> >>> In[2]:= Element[1.2, Reals] >>> >>> Out[2]= True > > the explanation is that Mathematica takes numbers written with a decimal > point and labels them "Real". This has nothing to do with their values, > which are, most assuredly, equal to rational numbers. And in > particular, 1.2==12/10 in Mathematica should trouble you if you believe > Mathematica speaks meaningfully on these issues. > >>> >>> and you might also read >>> >>> http://en.wikipedia.org/wiki/Computable_number >>> > But this would be irrelevant. > > The Mathematica documentation says, > "When domain membership cannot be decided the Element statement remain > [sic] unevaluated". > > "cannot be decided" > is not a statement about decidability in the technical "computability" > sense. It is a statement about this version of Mathematica not being > programmed to make a decision. Thus the fact that Mathematica 6.0 > cannot decide if e+pi is rational is not a deep result, and it is > referring to the mathematical literature about conjectures on the > matter. It just happens that the program fails to decide. The program > seems to be a jumble of some sort, since it knows that > > > > Mathematica 6.0 does not know e^e is definitely NOT Rational. (It is > known not to be rational). > > It just hasn't been programmed. Yet. It would be nice if the > documentation were clearer on this. > > Regards > RJF > > ..snip.. > -- DrMajorBob at yahoo.com