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Re: Re: Re: Re: algebraic numbers

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.


On Wed, 06 Jan 2010 04:56:38 -0600, Andrzej Kozlowski <akoz at>  

> 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

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