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

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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