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From: Noqsi on 5 Jan 2010 01:44 On Jan 4, 4:00 am, DrMajorBob <btre...(a)austin.rr.com> wrote: > > 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. > > So those are cultural conformity questions?!? One might not need to conform, but one must at least understand the culture. Mathematics is a human cultural artifact, and students are going to need to understand some things about that artifact and its expression to be successful in college. Specifically in this case series are often presented as specific terms and ellipsis, judged to be easier to comprehend in some ways than a formula, so the student should be able to comprehend that form. And this continues into professional life. Today I'm looking over the specs of a megapixel image sensor. The drawings that document its structure contain "..." in a number of places: it's not practical to show every pixel! I can, of course, think of all kinds of perverse and stupid ways to misunderstand what's omitted, but that wouldn't be helpful in any way. > > That's even worse than I thought! It's still worse. The intentions behind the widespread adoption of the SAT didn't really address the need to establish that the student could comprehend the academic cultural context: instead, they were consciously bigoted. http://www.newyorker.com/archive/2001/12/17/011217crat_atlarge > > Bobby > > > > On Sun, 03 Jan 2010 02:40:36 -0600, Noqsi <j...(a)noqsi.com> wrote: > > 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 unla= beled > >> 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 - (4= 37 (-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 sequen= ce > >> > can lead to further insight. > > >> -- > >> DrMajor...(a)yahoo.com > > -- > DrMajor...(a)yahoo.com
From: DrMajorBob on 5 Jan 2010 01:45 I didn't call it a formula. It's an algorithm, a very SIMPLE algorithm, and it always works. StringLength was never mentioned in the SATs, to my knowledge. Nor was any OTHER measure of simplicity. Nor was SIMPLICITY itself, for that matter. (I think.) Weren't questions posed as if there could only be one answer? Bobby On Mon, 04 Jan 2010 04:59:25 -0600, Andrzej Kozlowski <akoz(a)mimuw.edu.pl> wrote: > I am not sure I you can call that a "forumla". Besides > > StringLength["fill in the first bubble"] > > 24 > > So, if we also include them in the simplicity test, this one is probably > not going to pass. > > On 4 Jan 2010, at 07:38, DrMajorBob wrote: > >> The simplest algorithm for determining "what's the next term" is > simply "fill in the first bubble". >> >> Bobby >> >> On Sun, 03 Jan 2010 02:42:21 -0600, Andrzej Kozlowski > <akoz(a)mimuw.edu.pl> wrote: >> >>> >>> 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. >>> >>> >> >> >> -- >> DrMajorBob(a)yahoo.com > > -- DrMajorBob(a)yahoo.com
From: DrMajorBob on 5 Jan 2010 01:46 Computer reals are precisely equal to, and in one-to-one correspondence with, a miniscule subset of the rationals. Every one of them has a finite binary expansion. x = RandomReal[] digitForm = RealDigits@x; Length(a)First@digitForm rationalForm = FromDigits(a)digitForm {n, d} = Through[{Numerator, Denominator}@rationalForm] d x == n 0.217694 16 1088471616079187/5000000000000000 {1088471616079187, 5000000000000000} True A number can't get more rational or algebraic (solving a FIRST degree polynomial with integer coefficients) than that. If computer reals are THE reals, why is it that RandomReal[{3,4}] can never return Pi, Sqrt[11], or ANY irrational? OTOH, how often does RootApproximate(a)RandomReal[] succeed? Never, essentially: Reap(a)Do[x = RootApproximate(a)RandomReal[]; RootApproximate =!= Head@x && Sow@x, {10^8}] {Null, {}} Bobby On Mon, 04 Jan 2010 05:01:55 -0600, Noqsi <jpd(a)noqsi.com> wrote: > On Jan 3, 1:37 am, DrMajorBob <btre...(a)austin.rr.com> wrote: >> Mathematica Reals may not be Rational, but computer reals certainly are. >> (I shouldn't have capitalized "reals" in the second case.) > > Only in the shallow sense that there is a low entropy mapping between > computer "reals" and rational numbers in the intervals they represent. > But computer "reals" don't behave arithmetically like rationals or the > abstract "reals" of traditional mathematics. This fact often causes > confusion. > -- DrMajorBob(a)yahoo.com
From: DrMajorBob on 5 Jan 2010 01:47 Oops, big typo that time. I meant RootApproximant, not RootApproximate! With that corrected, I find that just about EVERY random real is algebraic. roots = First@ Last(a)Reap@ Do[x = RootApproximant(a)RandomReal[]; Root == Head@x && Sow@x, {10^2}]; Length(a)roots 100 (It's very slow, hence the small sample.) When RootApproximant fails, I suspect it's a limitation of the algorithm, not a property of the real. Bobby On Mon, 04 Jan 2010 15:17:56 -0600, DrMajorBob <btreat1(a)austin.rr.com> wrote: > Computer reals are precisely equal to, and in one-to-one correspondence > with, a miniscule subset of the rationals. Every one of them has a > finite binary expansion. > > x = RandomReal[] > digitForm = RealDigits@x; > Length(a)First@digitForm > rationalForm = FromDigits(a)digitForm > {n, d} = Through[{Numerator, Denominator}@rationalForm] > d x == n > > 0.217694 > > 16 > > 1088471616079187/5000000000000000 > > {1088471616079187, 5000000000000000} > > True > > A number can't get more rational or algebraic (solving a FIRST degree > polynomial with integer coefficients) than that. > > If computer reals are THE reals, why is it that RandomReal[{3,4}] can > never return Pi, Sqrt[11], or ANY irrational? > > OTOH, how often does RootApproximate(a)RandomReal[] succeed? > > Never, essentially: > > Reap(a)Do[x = RootApproximate(a)RandomReal[]; > RootApproximate =!= Head@x && Sow@x, {10^8}] > > {Null, {}} > > Bobby > > On Mon, 04 Jan 2010 05:01:55 -0600, Noqsi <jpd(a)noqsi.com> wrote: > >> On Jan 3, 1:37 am, DrMajorBob <btre...(a)austin.rr.com> wrote: >>> Mathematica Reals may not be Rational, but computer reals certainly >>> are. >>> (I shouldn't have capitalized "reals" in the second case.) >> >> Only in the shallow sense that there is a low entropy mapping between >> computer "reals" and rational numbers in the intervals they represent. >> But computer "reals" don't behave arithmetically like rationals or the >> abstract "reals" of traditional mathematics. This fact often causes >> confusion. >> > > -- DrMajorBob(a)yahoo.com
From: Andrzej Kozlowski on 6 Jan 2010 05:56
On 5 Jan 2010, at 15:47, DrMajorBob wrote: > If computer reals are THE reals, why is it that RandomReal[{3,4}] can > never return Pi, Sqrt[11], or ANY irrational? It can't possibly do that because these are computable real numbers the set of computable real numbers if countable and has measure 0. Computable numbers can never be the outcome of any distribution that selects numbers randomly from a real interval. The most common mistake people make about real numbers is to think that numbers such as Sqrt[2] or Pi as being in some sense typical examples of an irrational number or a transcendental number but they are not. They are very untypical because they are computable: that is, there exists a formula for computing as many of their digits as you like. But we can prove that the set of all reals with this property is countable and of measure 0. So Sqrt[2] is a very untypical irrational and Pi a very untypical transcendental. So what do typical real look like? Well, I think since a "typical" real is not computable we cannot know all of its digits and we cannot know any formula for computing them. But we can know a finite number of these digits. So this looks to me very much like the Mathematica concept of Real - you know a specified number of significant digits and you know that there are infinitely many more than you do not know. It seems to me the most natural way to think about non-computable reals. Roger Penrose, by the way, is famous for arguing that our brain is somehow able to work with non-computable quantities, although of course not by using digital expansions. But this involves quantum physics and has been the object of a heated dispute since the appearance of "The emperor's New Mind". = |