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Re: Re: Re: Re: algebraic numbers

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106280] Re: [mg106238] Re: [mg106220] Re: [mg106192] Re: algebraic numbers
  • From: DrMajorBob <btreat1 at austin.rr.com>
  • Date: Thu, 7 Jan 2010 02:30:28 -0500 (EST)
  • References: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net>
  • Reply-to: drmajorbob at yahoo.com

If I'm told that finite-precision reals are not Rational "because  
Mathematica says so", but that Mathematica success (by some algorithm) in  
finding a Root[...] representation doesn't mean the number is algebraic...  
yet I know that all finite binary expansions ARE both rational and  
algebraic as a matter of basic arithmetic... then I question whether  
Mathematica is saying anything either way.

Perhaps it's just Mathematica USERS holding forth in each direction.

I think the view of reals as monads (a la nonstandard analysis) melds with  
the fact that reals are irrational A.E. and non-algebraic A.E., while  
monads are, of course, consistent with the spirit of Mathematica's  
arbitrary-precision arithmetic (WHEN IT IS USED). The OP posted a number  
far beyond machine precision, so it's reasonable to come at this from that  
arbitrary-precision world-view... in which case you're "right" and I'm  
"wrong".

I called all the reals rational, and you called them monads (or  
equivalent).

Fine.

Bobby

On Wed, 06 Jan 2010 16:46:20 -0600, Andrzej Kozlowski <akoz at mimuw.edu.pl>  
wrote:

>
> On 7 Jan 2010, at 04:19, 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.
>
> No, they are not real algebraic. RootApproximant gives algenraic  
> approximations to these numbers and in fact it uses a test for what  
> makes a good approximation. In never says that these numbers themselves  
> are algebraic. You have been completely confused about this. The method  
> RootApproximant uses is the LLL method, which finds approximations.  
> Because of this it will give you a number of different approximations  
> for the same real. For example
>
> In[7]:= RootApproximant[N[Pi, 10], 2]
>
> Out[7]= (1/490)*(71 + Sqrt[2156141])
>
> In[8]:= RootApproximant[N[Pi, 10], 3]
>
> Out[8]= Root[37 #1^3-114 #1^2-36 #1+91&,3]
>
> So how come N[Pi,10] is equal to two quite different algebraic numbers?
> You should first understand what an algorithm does (e.g.  
> RootApproximant) before making weird claims about it. (In fact Daniel  
> Lichtblau already explained this but you just seem to have ignored it).
>
> Andrzej Kozlowski
>
>>
>> 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
>>
>> On Wed, 06 Jan 2010 04:57:26 -0600, Andrzej Kozlowski  
>> <akoz at mimuw.edu.pl> wrote:
>>
>>> Well, I think when you are using Mathematica it is the designers of
>>> Mathematica who decide what is rational and what is not.
>>>
>>> And when you are not using Mathematica (or other similar software which
>>> interprets certain computer data as numbers), than I can't imagine what
>>> you could possibly mean by a "computer number".
>>>
>>> Andrzej
>>>
>>>
>>> On 6 Jan 2010, at 11:45, DrMajorBob wrote:
>>>
>>>> Obviously, it DOES make them rational "in a sense"... the sense in
>>> which I mean it, for example.
>>>>
>>>> Bobby
>>>>
>>>> On Tue, 05 Jan 2010 20:41:34 -0600, Andrzej Kozlowski
>>> <akoz at mimuw.edu.pl> wrote:
>>>>
>>>>>
>>>>> On 6 Jan 2010, at 11:13, DrMajorBob wrote:
>>>>>
>>>>>> I completely understand that Mathematica considers 1.2 Real, not
>>> Rational... but that's a software design decision, not an objective
>>> fact.
>>>>>
>>>>> I think we are talking cross purposes. You seem to believe (correct
>>> me if I am wrong) that numbers somehow "exist". Well, I have never seen
>>> one - and that applies equally to irrational and rationals and even
>>> (contrary to Kronecker) integers. I do not know what the number 3 looks
>>> like, nor what 1/3 looks like (I know how we denote them, but that's  
>>> not
>>> the sam thing). So I do not think that the notion of "computer numbers"
>>> makes any sense and hence to say that all computer numbers are rational
>>> also does not make sense. There are only certain things that we
>>> interpret as numbers and when we interpret them as rationals they are
>>> rationals and when we interpret them as non-computable reals than they
>>> are just that.
>>>>> Of course we know that a computer can only store a finite number of
>>> such objects at a given time, but that fact in no sense makes them
>>> "rational".
>>>>>
>>>>> Andrzej Kozlowski
>>>>
>>>>
>>>> --
>>>> DrMajorBob at yahoo.com
>>>
>>>
>>
>>
>> --
>> DrMajorBob at yahoo.com
>


-- 
DrMajorBob at yahoo.com


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