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Re: Re: Re: algebraic numbers

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106234] Re: [mg106220] Re: [mg106192] Re: algebraic numbers
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Wed, 6 Jan 2010 05:56:38 -0500 (EST)
  • References: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net> <201001050647.BAA24123@smc.vnet.net>

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".

=


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