algebraic numbers
in [Mathematica]
|
Prev: FindInstance Problem
Next: Persistent assumption
From: Daniel Lichtblau on 8 Jan 2010 04:14 DrMajorBob wrote: >> Well, I think when you are using Mathematica it is the designers of >> Mathematica who decide what is rational and what is not. > > Not to repeat myself, but RootApproximant said 100 out of 100 randomly > chosen machine-precision reals ARE algebraic. > > If your interpretation is correct and consistent with Mathematica, and if > Mathematica is internally consistent on the topic, virtually all of those > reals should NOT have been algebraic. > > Mathematica designers wrote RootApproximant, I assume? > > Hence, I'd have to say your interpretation is no better than mine. > > Bobby Regarding RootApproximant design, the missing functionality is this. There is no limiting of coefficient size (or if there is, it's not obvious to me how it might be done). Rationalize has such limiting capability, more or less (though it is really built into the algorithm; the optional second argument does not impose it). A consequence is that all randoms can be made to fit some algebraic number of whatever degree, simply by allowing siufficiently large coefficients. I am not sure whether this is a design flaw. It might alternatively have been intentional, due to possible implementational difficulties in doing otherwise. In retrospect, it kinda surprises me that I am not familiar with the history of this particular design issue, but there you have it. Daniel Lichtblau Wolfram Research
From: Daniel Lichtblau on 8 Jan 2010 04:14 Richard Fateman wrote: > Andrzej Kozlowski wrote: >> [...] > 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. It is known whether this thread is rational. Empirical evidence seems to argue against it. > [...] > Maybe you think that Mathematica has a human mind? Of course she does. > (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. > ) That (a misfeature), or maybe it's a missing feature in some other programs. I rather like this behavior of Equal, though I agree there is good sense behind some recent criticisms to the effect that maybe it should be configurable (regarding bits of slop, or relative or absolute error specifications). > [...] > 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. I would be far more troubled if 1.2===12/10 gave True (that is, they were deemed SameQ rather than just Equal). Much of the town is shut down, including schools (though, alas, not the HS drama club trip). I had to shovel out this morning before work. I'll have to shovel again when I get home. So here I am, and it feels like I am still shovelling. Such sound and fury... Daniel Lichtblau Wolfram Research
From: Andrzej Kozlowski on 8 Jan 2010 04:14 On 7 Jan 2010, at 16:33, Richard Fateman wrote: > > Incidentally, it is not known if E+Pi is rational. Brilliant. But even a high school child can prove that either E+Pi is irrational or E*Pi is irrational (in fact you can replace irrational with transcendental). > > 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. You are confusing Real and Reals. This Element[1.2, Reals] True has nothing to do with types because Element[Pi, Reals] True even though Head /@ {1.2, Pi} {Real,Symbol} and also Element[0.5, Rationals] False Element[1/2, Rationals] True and just by the way: Element[Pi, Rationals] False Element[E, Rationals] False Element[E + Pi, Rationals] Element[E + Pi, Rationals] so Mathematica knows as much as you about this. But it does not know as much as me since Reduce[Element[E + Pi, Rationals] && Element[E Pi, Rationals]] should return False.
From: Richard Fateman on 8 Jan 2010 04:15 I agree with Bob. who says that "all computer reals are rationals" [The ones that are represented by fractions and exponents consisting of finite strings of bits.., not those that are symbols like Pi]. The fact that Penrose annoyed a lot of people with his attack on computer science does not mean he is right, or worthy of reading (though I have read his first opus). I tried another Google search and got .. Results 1 - 10 of about 188,000 for "hannah montana" +philosophy. (0.24 seconds) As for whether a computer program could conceptualize mathematics and "know" something, anyone could just say "no, computers can't". Others might point to programs that prove new theorems as a demonstration that computers "know math". (There are conferences on this topic, Mathematical Knowledge Management, and long-term research projects on theorem proving by computer). And a more cogent (but also deeply flawed) discussion of what computers can/can't do has been presented by John Searle, who is not a physicist, but a philosopher by profession. Others have said that computers will not replace mathematicians until computers learn to drink coffee. RJF
From: János Löbb on 8 Jan 2010 04:15
> Why can't a computer, in principle of > course, perfectly simulate the activity of the human brian that we > call > "doing mathematics"? Let me cite here an "alien from Mars" - Arthur K=F6stler : ) "The discoverer perceives relational patterns of functional analogies where nobody saw them before, as a poet perceives the image of a camel in a drifting cloud" or "The manner in which some of the most important individual discoveries were arrived at, remind one more of a sleepwalker's performance than an electronic brain's" J=E1nos= |