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Re: Re: Re: algebraic numbers

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106312] Re: [mg106272] Re: [mg106238] Re: algebraic numbers
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Fri, 8 Jan 2010 04:14:37 -0500 (EST)
  • References: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net> <201001070728.CAA23729@smc.vnet.net>

DrMajorBob wrote:
>> Well, I think when you are using Mathematica it is the designers of
>> Mathematica who decide what is rational and what is not.
> 
> Not to repeat myself, but RootApproximant said 100 out of 100 randomly  
> chosen machine-precision reals ARE algebraic.
> 
> If your interpretation is correct and consistent with Mathematica, and if  
> Mathematica is internally consistent on the topic, virtually all of those  
> reals should NOT have been algebraic.
> 
> Mathematica designers wrote RootApproximant, I assume?
> 
> Hence, I'd have to say your interpretation is no better than mine.
> 
> Bobby

Regarding RootApproximant design, the missing functionality is this. 
There is no limiting of coefficient size (or if there is, it's not 
obvious to me how it might be done). Rationalize has such limiting 
capability, more or less (though it is really built into the algorithm; 
the optional second argument does not impose it).

A consequence is that all randoms can be made to fit some algebraic 
number of whatever degree, simply by allowing siufficiently large 
coefficients.

I am not sure whether this is a design flaw. It might alternatively have 
been intentional, due to possible implementational difficulties in doing 
otherwise. In retrospect, it kinda surprises me that I am not familiar 
with the history of this particular design issue, but there you have it.

Daniel Lichtblau
Wolfram Research




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