Re: Re: Re: algebraic numbers

*To*: mathgroup at smc.vnet.net*Subject*: [mg106264] Re: [mg106220] Re: [mg106192] Re: algebraic numbers*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Wed, 6 Jan 2010 06:02:44 -0500 (EST)*References*: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net>*Reply-to*: drmajorbob at yahoo.com

I completely understand that Mathematica considers 1.2 Real, not Rational... but that's a software design decision, not an objective fact. If we consider something that's not representable in binary, it even makes a kind of sense: RealDigits[1/3.] {{3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3}, 0} But 1.2 _is_ representable in binary, that's the way it is represented in the computer, and there's no doubt about the digits, whatsoever: RealDigits[1.2] {{1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, 1} Even 1/3. and Sqrt[2.] are stored as members of a countable set of rationals... and representations of that sort are countable themselves (since they're all algebraic). But... I agree there's no point debating it. Bobby On Tue, 05 Jan 2010 16:04:20 -0600, Andrzej Kozlowski <akoz at mimuw.edu.pl> 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. > > 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 > > and you might also read > > http://en.wikipedia.org/wiki/Computable_number > > (but that's the last time I posting anything to do with any logic or > mathematics here.) > > Andrzej Kozlowski > > > > > > On 5 Jan 2010, at 22:31, DrMajorBob wrote: > >> RandomReal[] returns numbers from a countable set of rationals. >> >> Or call them reals, if you must; it still selects from a countable set >> of possibilities... not from the uncountable unit interval in the reals. >> >> The range of RandomReal[] is a set of measure zero, just like the >> algebraic numbers. >> >> Bobby >> >> On Tue, 05 Jan 2010 02:08:24 -0600, Andrzej Kozlowski >> <akoz at mimuw.edu.pl> wrote: >> >>> >>> On 5 Jan 2010, at 15:47, DrMajorBob wrote: >>> >>>> If computer reals are THE reals, why is it that RandomReal[{3,4}] can >>>> never return Pi, Sqrt[11], or ANY irrational? >>> >>> It can't possibly do that because these are computable real numbers >>> the set of computable real numbers if countable and has measure 0. >>> Computable numbers can never be the outcome of any distribution that >>> selects numbers randomly from a real interval. >>> >>> The most common mistake people make about real numbers is to think >>> that numbers such as Sqrt[2] or Pi as being in some sense typical >>> examples of an irrational number or a transcendental number but they >>> are not. They are very untypical because they are computable: that is, >>> there exists a formula for computing as many of their digits as you >>> like. But we can prove that the set of all reals with this property is >>> countable and of measure 0. So Sqrt[2] is a very untypical irrational >>> and Pi a very untypical transcendental. So what do typical real look >>> like? Well, I think since a "typical" real is not computable we cannot >>> know all of its digits and we cannot know any formula for computing >>> them. But we can know a finite number of these digits. So this looks >>> to me very much like the Mathematica concept of Real - you know a >>> specified number of significant digits and you know that there are >>> infinitely many more than you do not know. It seems to me the most >>> natural way to think about non-computable reals. >>> >>> Roger Penrose, by the way, is famous for arguing that our brain is >>> somehow able to work with non-computable quantities, although of >>> course not by using digital expansions. But this involves quantum >>> physics and has been the object of a heated dispute since the >>> appearance of "The emperor's New Mind". >>> >> >> >> -- >> DrMajorBob at yahoo.com > -- DrMajorBob at yahoo.com