algebraic numbers
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From: Andrzej Kozlowski on 2 Jan 2010 05:03 On 1 Jan 2010, at 19:36, DrMajorBob wrote: > "If so, you will indeed have recognized the number x as algebraic, from > its first N figures." > > No... you will have identified an algebraic number that agrees with x, to > N figures. > > OTOH, every computer Real is rational, so they're all algebraic. > > Bobby Well, actually Mathematica does not agree with your last assertion: Head[1.12] Real Element[1.12, Rationals] False Element[1.12, Reals] True In fact it seems clear that the designers of Mathematica have decided to interpret all approximate numbers (with head Real) as approximations to irrationals rather than as finite expansions of rationals. The only rationals in Mathematica are indeed the ones that have the head Rational, i.e. fractions. Andrzej Kozlowski > > On Thu, 31 Dec 2009 02:17:22 -0600, Robert Coquereaux > <robert.coquereaux(a)gmail.com> wrote: > >> "Impossible....Not at all" >> I think that one should be more precise: >> Assume that x algebraic, and suppose you know (only) its first 50 >> digits. Then consider y = x + Pi/10^100. >> Clearly x and y have the same first 50 digits , though y is not >> algebraic. >> Therefore you cannot recognize y as algebraic from its first 50 digits ! >> The quoted comment was in relation with the question first asked by >> hautot. >> Now, it is clear that, while looking for a solution x of some >> equation (or definite integral or...), one can use the answer obtained >> by applying RootApproximant (or another function based on similar >> algorithms) to numerical approximations of x, and then show that the >> suggested algebraic number indeed solves exactly the initial problem. >> If so, you will indeed have recognized the number x as algebraic, from >> its first N figures. >> But this does not seem to be the question first asked by hautot. >> Also, if one is able to obtain information, for any N, on the first N >> digits of a real number x, this is a different story... and a >> different question. >> >> Le 30 d=E9c. 2009 =E0 18:11, Daniel Lichtblau a =E9crit : >> >>> >>>> To recognize a number x as algebraic, from its N first figures, is >>>> impossible. >>> >>> Not at all. There are polynomial factorization algorithms based on >>> this notion (maybe you knew that). >>> >>> Daniel Lichtblau >>> Wolfram Research >> >> > > > -- > DrMajorBob(a)yahoo.com >
From: Noqsi on 3 Jan 2010 03:37 On Jan 2, 3:05 am, DrMajorBob <btre...(a)austin.rr.com> wrote: > When I clicked on the link below, the search field was already filled with > the sequence > > target = {1, 2, 3, 6, 11, 23, 47, 106, 235}; > > Searching yielded "A000055 Number of trees with n unlabeled nodes." > > I tried a few Mathematica functions on it: > > FindLinearRecurrence(a)target > > FindLinearRecurrence[{1, 2, 3, 6, 11, 23, 47, 106, 235}] > > (fail) > > FindSequenceFunction(a)target > > FindSequenceFunction[{1, 2, 3, 6, 11, 23, 47, 106, 235}] > > (fail) > > f[x_] = InterpolatingPolynomial[target, x] > > 1 + (1 + (1/ > 3 + (-(1/ > 12) + (7/ > 120 + (-(1/ > 60) + (1/144 - (41 (-8 + x))/20160) (-7 + x)) (-6 + > x)) (-5 + x)) (-4 + x)) (-3 + x) (-2 + x)) (-1 + x) > > and now the next term: > > Array[f, 1 + Length(a)target] > > {1, 2, 3, 6, 11, 23, 47, 106, 235, 322} > > But, unsurprisingly, the next term in A000055 is 551, not 322. > > A000055 actually starts with another three 1s, but that doesn't change > things much: > > target = {1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235}; > > FindLinearRecurrence(a)target > > FindLinearRecurrence[{1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235}] > > (fail) > > FindSequenceFunction(a)target > > FindSequenceFunction[{1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235}] > > (fail) > > f[x_] = InterpolatingPolynomial[target, x] > > 1 + (1/24 + (-(1/ > 40) + (1/ > 90 + (-(1/ > 280) + (1/ > 1008 + (-(43/ > 181440) + (191/3628800 - (437 (-11 + x))/ > 39916800) (-10 + x)) (-9 + x)) (-8 + x)) (-7 + > x)) (-6 + x)) (-5 + x)) (-4 + x) (-3 + x) (-2 + x) (-1 + > x) > > Array[f, 1 + Length(a)target] > > {1, 1, 1, 1, 2, 3, 6, 11, 23, 47, 106, 235, -502} > > So I ask you, from the data alone: what's the next term? It's the sort of question where one might expect a specialist to recognize a familiar sequence. It's all context. Consider that in a narrow mathematical sense, spectroscopy is an utterly ambiguous, "ill conditioned" problem. But show me a gigagauss cyclotron spectrum, and I'll recognize it as such (see the acknowledgment at the end of arxiv.org/pdf/astro-ph/0306189: the authors were struggling to contrive an interpretation from atomic physics before one of them showed the spectrum to me). But I expect very few could do this, since few have the background. > > If one had the Encyclopedia of Integer Sequences handy, those SAT > questions could be interesting. But they'd still be nonsense. No they are not. Remember that the SAT isn't about the ability of a student to function in some ideal abstract world of infinite possibility. In the real world of academia, every single question they will encounter will be ambiguous in some sense. The issue here is whether the student has enough common culture with the test writer to find the same answer. And that's *always* an issue. > > Bobby > > > > On Fri, 01 Jan 2010 04:32:58 -0600, Noqsi <j...(a)noqsi.com> wrote: > > On Dec 31, 1:16 am, DrMajorBob <btre...(a)austin.rr.com> wrote: > > >> This is a little like those idiotic SAT and GRE questions that ask > >> "What's > >> the next number in the following series?"... where any number will do. > >> Test writers don't seem to know there's an interpolating polynomial (for > >> instance) to fit the given series with ANY next element. > > > Explanations in terms of epicycles may be mathematically adequate in a > > narrow sense, but an explanation in terms of a single principle > > applied repeatedly is to be preferred in science. The ability to > > recognize such a principle is important. > > > And my mathematical logician son (who's looking over my shoulder) > > directed me tohttp://www.research.att.com/~njas/sequences/for > > research on this topic. When he encounters such a sequence in his > > research, he finds that knowledge of a simple genesis for the sequence > > can lead to further insight. > > -- > DrMajor...(a)yahoo.com
From: DrMajorBob on 3 Jan 2010 03:37 Mathematica Reals may not be Rational, but computer reals certainly are. (I shouldn't have capitalized "reals" in the second case.) Bobby On Sat, 02 Jan 2010 04:03:30 -0600, Andrzej Kozlowski <akoz(a)mimuw.edu.pl> wrote: > > On 1 Jan 2010, at 19:36, DrMajorBob wrote: > >> "If so, you will indeed have recognized the number x as algebraic, from >> its first N figures." >> >> No... you will have identified an algebraic number that agrees with x, >> to >> N figures. >> >> OTOH, every computer Real is rational, so they're all algebraic. >> >> Bobby > > Well, actually Mathematica does not agree with your last assertion: > > Head[1.12] > > Real > > Element[1.12, Rationals] > > False > > Element[1.12, Reals] > > True > > In fact it seems clear that the designers of Mathematica have decided to > interpret all approximate numbers (with head Real) as approximations to > irrationals rather than as finite expansions of rationals. The only > rationals in Mathematica are indeed the ones that have the head > Rational, i.e. fractions. > > Andrzej Kozlowski > > > >> >> On Thu, 31 Dec 2009 02:17:22 -0600, Robert Coquereaux >> <robert.coquereaux(a)gmail.com> wrote: >> >>> "Impossible....Not at all" >>> I think that one should be more precise: >>> Assume that x algebraic, and suppose you know (only) its first 50 >>> digits. Then consider y = x + Pi/10^100. >>> Clearly x and y have the same first 50 digits , though y is not >>> algebraic. >>> Therefore you cannot recognize y as algebraic from its first 50 digits >>> ! >>> The quoted comment was in relation with the question first asked by >>> hautot. >>> Now, it is clear that, while looking for a solution x of some >>> equation (or definite integral or...), one can use the answer obtained >>> by applying RootApproximant (or another function based on similar >>> algorithms) to numerical approximations of x, and then show that the >>> suggested algebraic number indeed solves exactly the initial problem. >>> If so, you will indeed have recognized the number x as algebraic, from >>> its first N figures. >>> But this does not seem to be the question first asked by hautot. >>> Also, if one is able to obtain information, for any N, on the first N >>> digits of a real number x, this is a different story... and a >>> different question. >>> >>> Le 30 d=E9c. 2009 =E0 18:11, Daniel Lichtblau a =E9crit : >>> >>>> >>>>> To recognize a number x as algebraic, from its N first figures, is >>>>> impossible. >>>> >>>> Not at all. There are polynomial factorization algorithms based on >>>> this notion (maybe you knew that). >>>> >>>> Daniel Lichtblau >>>> Wolfram Research >>> >>> >> >> >> -- >> DrMajorBob(a)yahoo.com >> > > -- DrMajorBob(a)yahoo.com
From: Andrzej Kozlowski on 3 Jan 2010 03:38 Actually, in Mathematica one has to carefully distinguish between Real and Reals, Rational and Rationals, Integer and Integers etc. They are really very different entities Mathematica Reals contain the Rationals. Element[2/3, Reals] True So a Rational belongs to the Reals but is not of the type Real.Also Element[2/3, Rationals] True Element[1.2, Rationals] False I understand that what you mean is something completely different - because Mathematica approximate numbers have only finite digits they are "rationals". But this is really only a matter of how you choose to interpret them: Mathematica interprets them as the first digits of irrational numbers whose remaining digits are unknown (but there are always infinitely many of them and they do not repeat themselves). This is quite a consistent approach and in fact the only one that justifies using approximate numbers in continuous probability distributions. Philosophically speaking, we cannot be sure that that is not exactly what we do in our own minds when we thing of "real numbers" - our brains are quite possibly only finite state automata and our reals are also approximations to "reals" most of whose digits always remain unknown. So I am not convinced that there is in this respect any fundamental difference between humans and computers. And by the way, reals numbers such as Pi or Sqrt[2], for which you have a method which computes their digits as far as you wish to do are also countable and form a set of measure zero. Most of real numbers are not computable, either by computers or by us. On 3 Jan 2010, at 06:34, DrMajorBob wrote: > Mathematica Reals may not be Rational, but computer reals certainly are. (I shouldn't have capitalized "reals" in the second case.) > > Bobby > > On Sat, 02 Jan 2010 04:03:30 -0600, Andrzej Kozlowski <akoz(a)mimuw.edu.pl> wrote: > >> >> On 1 Jan 2010, at 19:36, DrMajorBob wrote: >> >>> "If so, you will indeed have recognized the number x as algebraic, from >>> its first N figures." >>> >>> No... you will have identified an algebraic number that agrees with x, to >>> N figures. >>> >>> OTOH, every computer Real is rational, so they're all algebraic. >>> >>> Bobby >> >> Well, actually Mathematica does not agree with your last assertion: >> >> Head[1.12] >> >> Real >> >> Element[1.12, Rationals] >> >> False >> >> Element[1.12, Reals] >> >> True >> >> In fact it seems clear that the designers of Mathematica have decided to interpret all approximate numbers (with head Real) as approximations to irrationals rather than as finite expansions of rationals. The only rationals in Mathematica are indeed the ones that have the head Rational, i.e. fractions. >> >> Andrzej Kozlowski >> >> >> >>> >>> On Thu, 31 Dec 2009 02:17:22 -0600, Robert Coquereaux >>> <robert.coquereaux(a)gmail.com> wrote: >>> >>>> "Impossible....Not at all" >>>> I think that one should be more precise: >>>> Assume that x algebraic, and suppose you know (only) its first 50 >>>> digits. Then consider y = x + Pi/10^100. >>>> Clearly x and y have the same first 50 digits , though y is not >>>> algebraic. >>>> Therefore you cannot recognize y as algebraic from its first 50 digits ! >>>> The quoted comment was in relation with the question first asked by >>>> hautot. >>>> Now, it is clear that, while looking for a solution x of some >>>> equation (or definite integral or...), one can use the answer obtained >>>> by applying RootApproximant (or another function based on similar >>>> algorithms) to numerical approximations of x, and then show that the >>>> suggested algebraic number indeed solves exactly the initial problem. >>>> If so, you will indeed have recognized the number x as algebraic, from >>>> its first N figures. >>>> But this does not seem to be the question first asked by hautot. >>>> Also, if one is able to obtain information, for any N, on the first N >>>> digits of a real number x, this is a different story... and a >>>> different question. >>>> >>>> Le 30 d=E9c. 2009 =E0 18:11, Daniel Lichtblau a =E9crit : >>>> >>>>> >>>>>> To recognize a number x as algebraic, from its N first figures, is >>>>>> impossible. >>>>> >>>>> Not at all. There are polynomial factorization algorithms based on >>>>> this notion (maybe you knew that). >>>>> >>>>> Daniel Lichtblau >>>>> Wolfram Research >>>> >>>> >>> >>> >>> -- >>> DrMajorBob(a)yahoo.com >>> >> >> > > > -- > DrMajorBob(a)yahoo.com
From: Andrzej Kozlowski on 3 Jan 2010 03:38
On 2 Jan 2010, at 19:05, DrMajorBob wrote: > If one had the Encyclopedia of Integer Sequences handy, those SAT > questions could be interesting. But they'd still be nonsense. I think only their wording should be changed to something like: "find the simplest formula generating the sequence ... where simplest means requiring the least number of standard mathematical symbols to write down". These kind of questions, of course, should probably allow the possibility that the person answering the question finds an answer that is simpler than the expected one. |