Re: Re: Re: algebraic numbers
- To: mathgroup at smc.vnet.net
- Subject: [mg106265] Re: [mg106220] Re: [mg106192] Re: algebraic numbers
- From: DrMajorBob <btreat1 at austin.rr.com>
- Date: Wed, 6 Jan 2010 06:02:55 -0500 (EST)
- References: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net>
- Reply-to: drmajorbob at yahoo.com
> is weird - there is no such things as "computer numbers". Numbers exist > only and (probably) exclusively in the human mind. Well... computer reals (floating point numbers) certainly DO exist, and that's the way Mathematica stores a (machine precision) real number. I never said MATHEMATICA Reals were all algebraic, mind you... because your point is well taken that Mathematica has an interpretation or "point of view" that may conflict with mine. Even so... when Mathematica is asked the question via RootApproximant, I haven't located a machine-precision number that isn't algebraic. When we go beyond machine precision, your point is far more solid... but I suspect a sufficiently stubborn and exhaustive RootApproximant would do the same with those. Bobby On Tue, 05 Jan 2010 16:39:50 -0600, Andrzej Kozlowski <akoz at mimuw.edu.pl> 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. > 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. Mathematica interprets them as > non-computable irrationals in order to make mathematical sense when > returning them while simulation a real distribution, because all other > numbers have measure 0. > 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. > > 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. > > 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. >> >> 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