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Re: Re: algebraic numbers
*To*: mathgroup at smc.vnet.net
*Subject*: [mg106323] Re: [mg106295] Re: algebraic numbers
*From*: DrMajorBob <btreat1 at austin.rr.com>
*Date*: Fri, 8 Jan 2010 04:16:42 -0500 (EST)
*References*: <200912290620.BAA02732@smc.vnet.net> <hhpl0g$9l1$1@smc.vnet.net>
*Reply-to*: drmajorbob at yahoo.com
> And in
> particular, 1.2==12/10 in Mathematica should trouble you if you believe
> Mathematica speaks meaningfully on these issues.
That's it, in a nutshell.
Bobby
On Thu, 07 Jan 2010 01:33:33 -0600, Richard Fateman
<fateman at cs.berkeley.edu> wrote:
> Andrzej Kozlowski wrote:
>> Just one more comment, I hope my last one on this subject. Obviously
>> RandomReal make it choices out of a countable set of entities. One would
>> have to be insane to claim otherwise and I am not that yet.
>>
>> But, Mathematica does not regard these entities as rational numbers and
>> so they are not that. If you call them rationals the it does not make
>> *mathematical* sense (because rationals have measure 0). So, if
>> Mathemaitca does not regard them as rationals they are not rationals.
>
> You can refer to Rationals as whatever Mathematica
> calls Rationals. But the rational numbers include all numbers that are
> represented by finite explicit binary strings in a floating-point
> format. They also include other numbers whose binary expansions are
> infinite, but repeat.
>
> Can Mathematica represent Reals that are NOT RATIONAL? Sure. Here are
> examples: Sqrt[2], 3*Pi, 4*E. 3*E +4*E^E + 5*E^E^E.
> Incidentally, it is not known if E+Pi is rational.
>
>> How could they be that ? Until they are interpreted by Mathematica, they
>> are not numbers at all but just some data stored in computer memory -
>> which are not numbers of any kind.
> true, but other programs can also interpret them. As numbers, as ASCII
> character strings, as pointers into memory.
> Mathematica interprets them as
>> non-computable irrationals
>
> No, that's not the way computer programs work. Mathematica allows some
> set of operations like +, *, printing. That's all. They are obviously
> computable and finitely representable as well, but Mathematica doesn't
> need to have an opinion on this, and neither do we have to attribute
> opinions to Mathematica.
>
> If you think that the operations that Mathematica performs are
> consistent with YOUR view that these numbers are non-computable
> irrationals, I suppose that is your view, but it is certainly
> unnecessary for others to hold this view.
>
> in order to make mathematical sense when
>> returning them while simulation a real distribution, because all other
>> numbers have measure 0.
>
> There is a literature on pseudo-random numbers that makes mathematical
> sense without any such interpretation.
>>
>> This is all about "simulating mathematics" - numbers do not live in any
>> sense inside computers. To say that "all computer numbers are rational"
>> is weird - there is no such things as "computer numbers". Numbers exist
>> only and (probably) exclusively in the human mind.
>
> Actually, you just said that Mathematica interprets --blah blah. Maybe
> you think that Mathematica has a human mind?
>>
>> To say that 1.2 is rational in Mathematica even if Mathematica says
>>
>> Element[1.2, Rationals]
>>
>> False
>>
>> does not make any sense at all.
>
> It makes perfect sense to say (in Mathematica) that 1.2 is a rational
> number because it is equal to a rational number. Huh??
> 1.2==12/10
> True.
>
> (A better example would be 1.25, since 1.2 is not representable exactly
> in binary. This example of 1.2 actually reveals a "misfeature of
> mathematica.
>
> 1.2==5404319552844595/4503599627370496
> True.
>
> So 1.2 is actually Mathematica-equal to another rational number. Many,
> in fact.
> )
>
>
> If you capitalize the term and wish to say that 1.2 is not a Rational
> in Mathematica, that is just a convention based on the "type" of data
> that is input to Mathematica with a decimal point and is therefore
> stored in a memory format that is labeled "Real" which (in Mathematica)
> is a superclass of "Rational". That is, "1/2" is a Rational but is also
> a Real and is incidentally also a Complex. But 0.5 is not a Real.
>
> This categorization of types in Mathematica does not determine the
> membership (or not) of a particular numeric VALUE in a mathematical
> category such as "rational". From a mathematical perspective, any
> legal number in an ordinary floating-point format can represent only a
> rational value.
>
>>
>> Andrzej Kozlowski
>>
>>
>>
>>
>> On 6 Jan 2010, at 07:04, Andrzej Kozlowski wrote:
>>
>>> Well, you are obviously misunderstanding what I am trying to explain
>> but I have no desire to spend any more time on it. I give up.
>
> OK.
>
>>> Perhaps you should try to explain yourself why Mathematica gives
>>>
>>> In[1]:= Element[1.2, Rationals]
>>>
>>> Out[1]= False
>>>
>>> In[2]:= Element[1.2, Reals]
>>>
>>> Out[2]= True
>
> the explanation is that Mathematica takes numbers written with a decimal
> point and labels them "Real". This has nothing to do with their values,
> which are, most assuredly, equal to rational numbers. And in
> particular, 1.2==12/10 in Mathematica should trouble you if you believe
> Mathematica speaks meaningfully on these issues.
>
>>>
>>> and you might also read
>>>
>>> http://en.wikipedia.org/wiki/Computable_number
>>>
> But this would be irrelevant.
>
> The Mathematica documentation says,
> "When domain membership cannot be decided the Element statement remain
> [sic] unevaluated".
>
> "cannot be decided"
> is not a statement about decidability in the technical "computability"
> sense. It is a statement about this version of Mathematica not being
> programmed to make a decision. Thus the fact that Mathematica 6.0
> cannot decide if e+pi is rational is not a deep result, and it is
> referring to the mathematical literature about conjectures on the
> matter. It just happens that the program fails to decide. The program
> seems to be a jumble of some sort, since it knows that
>
>
>
> Mathematica 6.0 does not know e^e is definitely NOT Rational. (It is
> known not to be rational).
>
> It just hasn't been programmed. Yet. It would be nice if the
> documentation were clearer on this.
>
> Regards
> RJF
>
> ..snip..
>
--
DrMajorBob at yahoo.com
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