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Re: Re: Re: Re: algebraic numbers

• To: mathgroup at smc.vnet.net
• Subject: [mg106279] Re: [mg106238] Re: [mg106220] Re: [mg106192] Re: algebraic numbers
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Thu, 7 Jan 2010 02:30:15 -0500 (EST)
• References: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net> <201001050647.BAA24123@smc.vnet.net> <E44EA2F2-1274-43E8-93DE-DC5BD31884A5@mimuw.edu.pl> <op.u52ai6jwtgfoz2@bobbys-imac.local> <504E0A05-61DB-4A43-9637-68216076623C@mimuw.edu.pl> <op.u529salwtgfoz2@bobbys-imac.local> <771DE886-36BB-4108-A83C-808109BAA8C3@mimuw.edu.pl> <op.u53a91u1tgfoz2@bobbys-imac.local> <201001061057.FAA14928@smc.vnet.net> <op.u54k94ojtgfoz2@bobbys-imac.local>

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 algebraic 
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

• References:
  • Re: Re: algebraic numbers
    • From: DrMajorBob <btreat1@austin.rr.com>
  • Re: Re: Re: algebraic numbers
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>