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MathGroup Archive 2005

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Re: Re: Re: Re: Re: Types in Mathematica thread


This is fascinating but due to relevancy we can't pursue more here;
except to more precisely identify what Mathematica knows. From Help:

"Root objects are the way that Mathematica represents algebraic
numbers. Algebraic numbers have the property that when you perform
algebraic operations on them, you always get a single algebraic number
as the result." 3.4.3

"Algebraic numbers are defined to be numbers that solve polynomial
equations with rational coefficients.

x (* is an element of *) Algebraics evaluates immediately only for
quantities x that are explicitly constructed from rational numbers,
radicals and Root objects, or are known to be transcendental." (* from
the Built-in Function definition of Algebraics *)

So what does it do, try to solve the expression with Simplify, Root,
RootReduce, & other functions, and then, if those functions don't
work, check a look up table for known transcendentals? But is it
reliable--if it can't solve it, how does it know it's transcendental?
Interestingly:

In[91]:= Not[Element[2^(1/3)+Sqrt[3],Algebraics]]


Out[91]= False

How does it know that it's not a transcendental when it can't solve
for roots? The following is from Help 3.4.3 about the expression just
tested:

"Even though a simple form in terms of radicals does exist, ToRadicals
does not find it."

How does it know, then?

"However, you should realize that there are some special cases in
which a reduction to radicals is in principle possible, but
Mathematica cannot find it."

Lastly: any recommended math user groups?

Kris

On 12/6/05, Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
>
> On 6 Dec 2005, at 14:03, Kristen W Carlson wrote:
>
> > You had me going there, I did look for it :-)
> >
> > Maybe. Another possibility is the ambiguity; an integer, a negative
> > number, a rational, a prime, are all reals.
>
> Yes, but obviously if the discussion is about types in the sense that
> computer scientists (not me!) use the term, then the type of objects
> with Head Real in Mathematica is exactly what is known as inexact
> numbers, or floating point number or floats etc..
>

> > Andrzej, why there isn't a test for transcendentals--TranscendentalQ.
> > That's a joke but I always wanted to see how pi and E were proved to
> > be transcendental. I wonder if there is an algorithm to capture some
> > of them, classes of them or something. Those proofs, since they are
> > finite, must capture some commonality.
>
> Well, actually the proofs for E and Pi are quite different. The proof
> of the transcendentality of E was given by Hermite and that of Pi by
> Lindemann about a decade later (both in the 19th century).  Actually,
> it is quite easy to prove that E is transcendental  (a simple proof
> was given by Hilbert). Hilbert also gave a proof of the fact that Pi
> is transcendental, which is simpler than Lindemann's original one,
> but still much harder than the proof for Pi. Many years ago as an
> undergraduate student I saw Hilbert's proof of the trancendentality
> of E done in a number theory class but all I remember now is that you
> start by assuming that there is a polynomial with integer
> coefficients whose root is E, and than you use properties of some
> infinite integrals involving E to prove that this is impossible. I
> have no idea how one proves that Pi is transcendental, but I suppose
> the general scheme must be similar. It is much easier to prove that
> there are transcendental numbers (actually, if you accept Cantor's
> diagonal argument - Kronecker did not - then it is almost trivial).
> In fact one can also construct explicit numbers for which it is easy
> to prove that they are transcendental: the first one was constructed
> by Liouville. This number, the Liouville constant, is
>
> Sum[10^(-k!),{k,1,Infinity}]
>
> Mathematica does not know it but correctly computes its numerical
> approximation:
>
>
> NSum[10^(-k!), {k, 1, Infinity}]
>
> 0.110001
>
> But in any case, all this has nothing to do with "types" in the sense
> of computer science.
>
> Andrzej Kozlowski
>
> >
> > Kris
> >
> > On 12/5/05, Andrzej Kozlowski <akoz at mimuw.edu.pl> wrote:
> >>
> >> On 5 Dec 2005, at 17:37, Kristen W Carlson wrote:
> >>
> >>> I can't think of why there is no RealQ predicate, but there is
> >>> _Real,
> >>> a pattern test via the head.
> >>>
> >>
> >> Maybe because it is called InexactNumberQ.
> >>
> >> Andrzej Kozlowski
> >>
> >>
> >
>
>


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