Re: Re: Re: Re: algebraic numbers

*To*: mathgroup at smc.vnet.net*Subject*: [mg106299] Re: [mg106234] Re: [mg106220] Re: [mg106192] Re: algebraic numbers*From*: DrMajorBob <btreat1 at austin.rr.com>*Date*: Thu, 7 Jan 2010 02:34:22 -0500 (EST)*References*: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net>*Reply-to*: drmajorbob at yahoo.com

We've already noted that RandomReal[] outputs are not only of measure zero... but also finite. Hence, arguing that computable numbers have measure zero and hence can't be RandomReal[] outputs, seems beside the point. Bobby On Wed, 06 Jan 2010 04:56:38 -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