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

*To*: mathgroup at smc.vnet.net*Subject*: [mg62869] Re: [mg62843] Re: [mg62824] Re: [mg62787] Re: [mg62781] Re: [mg62708] Types in Mathematica thread*From*: Kristen W Carlson <carlsonkw at gmail.com>*Date*: Tue, 6 Dec 2005 23:11:24 -0500 (EST)*References*: <dmp9na$hi2$1@smc.vnet.net> <roadnYOk3NcFDw7eRVn-jg@speakeasy.net> <200512050837.DAA08425@smc.vnet.net> <200512051840.NAA21063@smc.vnet.net> <200512060503.AAA02736@smc.vnet.net> <200512060858.DAA09068@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

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 > >> > >> > > > >

**References**:**Re: Types in Mathematica thread***From:*Kristen W Carlson <carlsonkw@gmail.com>

**Re: Re: Types in Mathematica thread***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>

**Re: Re: Re: Types in Mathematica thread***From:*Kristen W Carlson <carlsonkw@Gmail.com>

**Re: Re: Re: Re: Types in Mathematica thread***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>