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Re: Re: Re: algebraic numbers

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106245] Re: [mg106220] Re: [mg106192] Re: algebraic numbers
  • From: DrMajorBob <btreat1 at austin.rr.com>
  • Date: Wed, 6 Jan 2010 05:58:51 -0500 (EST)
  • References: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net>
  • Reply-to: drmajorbob at yahoo.com

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


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