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Re: Re: Re: algebraic numbers

• To: mathgroup at smc.vnet.net
• Subject: [mg106327] Re: [mg106272] Re: [mg106238] Re: algebraic numbers
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Fri, 8 Jan 2010 04:17:28 -0500 (EST)
• References: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net>
• Reply-to: drmajorbob at yahoo.com

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

And sufficiently large degree? A combination of the two?

Bobby

On Thu, 07 Jan 2010 10:19:39 -0600, Daniel Lichtblau <danl at wolfram.com>  
wrote:

> 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
>
>


-- 
DrMajorBob at yahoo.com