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Re: Re: Creating a Random Function to Select an Irrational

  • To: mathgroup at smc.vnet.net
  • Subject: [mg102244] Re: [mg102210] Re: [mg102176] Creating a Random Function to Select an Irrational
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Sun, 2 Aug 2009 06:01:17 -0400 (EDT)
  • References: <200907310957.FAA19545@smc.vnet.net> <200908010801.EAA07130@smc.vnet.net> <2B9B5B52-EA39-418D-AE5F-269570E18AC0@mimuw.edu.pl>

On 1 Aug 2009, at 20:54, Andrzej Kozlowski wrote:

> It may be more of a philosophical than mathematical issue, but I do  
> not think it is correct to say that approximate numbers are  
> rational. This is confirmed by Mathematica itself:
>
> Element[1.1, Rationals]
> False
>
> This is not just some kind of Mathematica weirdness but a perfectly  
> logical interpretation of approximate numbers. They are neither  
> rational not irrational; any one of them represents infinitely many  
> possible rational or irrational numbers. If you chose to interpret  
> approximate numbers as rational then you could not possibly claim  
> that RandomReal[{0,1}] is uniformly distributed on the unit interval  
> because there are infinitely many rationals e.g. 1/3 that can never  
> be obtained as the value of RandomReal. So the only sensible  
> interpretation is that all approximate numbers are neither rational  
> nor irrational - we can think of them as given by "initial digits"   
> of rational or irrational reals.


Another reasonable interpretation is that Mathematica's approximate  
numbers are actually (approximations) of irrationals. This is  
reasonable in view of the fact that RandomReal is supposed to return  
uniformly distributed reals - as you correctly pointed out the  
probability of obtaining a rational is 0. Hence it is reasonable to  
adopt the interpretation that each Mathematica Real stand actually for  
infinitely many irrationals - all with the same initial digits,  
depending on the precision.

Andrzej Kozlowski


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