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Re: algebraic numbers


Actually, in Mathematica one has to carefully distinguish between Real 
and Reals, Rational and Rationals, Integer and Integers etc. They are 
really very different entities

Mathematica Reals contain the Rationals.

Element[2/3, Reals]

 True

So a Rational belongs to the Reals but is not of the type Real.Also

 Element[2/3, Rationals]

True

Element[1.2, Rationals]

False

I understand that what you mean is something completely different - 
because Mathematica approximate numbers have only finite digits they are 
"rationals". But this is really only a matter of how you choose to 
interpret them: Mathematica interprets them as the first digits of 
irrational numbers whose remaining digits are unknown (but there are 
always infinitely many of them and they do not repeat themselves).

This is quite a consistent approach and in fact the only one that 
justifies using approximate numbers in continuous probability 
distributions.

Philosophically speaking, we cannot be sure that that is not exactly 
what we do in our own minds when we thing of "real numbers" - our brains 
are quite possibly only finite state automata and our reals are also 
approximations to "reals" most of whose digits always remain unknown. So 
I am not convinced that there is in this respect any fundamental 
difference between humans and computers.

And by the way, reals numbers such as Pi or Sqrt[2], for which you have 
a method which computes their digits as far as you wish to do are also 
countable and form a set of measure zero. Most of real numbers are not 
computable, either by computers or by us.



On 3 Jan 2010, at 06:34, DrMajorBob wrote:

> Mathematica Reals may not be Rational, but computer reals certainly 
are. (I shouldn't have capitalized "reals" in the second case.)
>
> Bobby
>
> On Sat, 02 Jan 2010 04:03:30 -0600, Andrzej Kozlowski 
<akoz at mimuw.edu.pl> wrote:
>
>>
>> On 1 Jan 2010, at 19:36, DrMajorBob wrote:
>>
>>> "If so, you will indeed have recognized the number x as algebraic, 
from
>>> its first N figures."
>>>
>>> No... you will have identified an algebraic number that agrees with 
x, to
>>> N figures.
>>>
>>> OTOH, every computer Real is rational, so they're all algebraic.
>>>
>>> Bobby
>>
>> Well, actually Mathematica does not agree with your last assertion:
>>
>> Head[1.12]
>>
>> Real
>>
>> Element[1.12, Rationals]
>>
>> False
>>
>> Element[1.12, Reals]
>>
>> True
>>
>> In fact it seems clear that the designers of Mathematica have decided 
to interpret all approximate numbers (with head Real) as approximations 
to irrationals rather than as finite expansions of rationals. The only 
rationals in Mathematica are indeed the ones that have the head 
Rational, i.e. fractions.
>>
>> Andrzej Kozlowski
>>
>>
>>
>>>
>>> On Thu, 31 Dec 2009 02:17:22 -0600, Robert Coquereaux
>>> <robert.coquereaux at gmail.com> wrote:
>>>
>>>> "Impossible....Not at all"
>>>> I think that one should be more precise:
>>>> Assume that x algebraic, and suppose you know (only) its first 50
>>>> digits. Then consider y = x +  Pi/10^100.
>>>> Clearly x and y have the same first 50 digits , though y is not
>>>> algebraic.
>>>> Therefore you cannot recognize y as algebraic from its first 50 
digits !
>>>> The quoted comment was in relation with the question first asked by
>>>> hautot.
>>>> Now, it is clear that, while looking for a solution x of  some
>>>> equation (or definite integral or...), one can use the answer 
obtained
>>>> by applying  RootApproximant (or another function based on similar
>>>> algorithms) to numerical approximations of x, and then show that 
the
>>>> suggested algebraic number indeed solves exactly the initial 
problem.
>>>> If so, you will indeed have recognized the number x as algebraic, 
from
>>>> its first N figures.
>>>> But this does not seem to be the question first asked by hautot.
>>>> Also, if one is able to obtain information, for any N, on the first 
N
>>>> digits of a real number x, this is a different story... and a
>>>> different question.
>>>>
>>>> Le 30 d=E9c. 2009 =E0 18:11, Daniel Lichtblau a =E9crit :
>>>>
>>>>>
>>>>>> To recognize a number x as algebraic, from its N first figures, 
is
>>>>>> impossible.
>>>>>
>>>>> Not at all. There are polynomial factorization algorithms based on
>>>>> this notion (maybe you knew that).
>>>>>
>>>>> Daniel Lichtblau
>>>>> Wolfram Research
>>>>
>>>>
>>>
>>>
>>> --
>>> DrMajorBob at yahoo.com
>>>
>>
>>
>
>
> --
> DrMajorBob at yahoo.com



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