July 2003 - mathfun@mailman.xmission.com

[math-fun] Is Z^{>=} a rig?
by Mike Stay 03 Jul '03

03 Jul '03

I haven't been following the thread very closely, but I noticed you were trying to come up with a name for something defined over the naturals. Does this definition fit your structure? http://www.wikipedia.org/wiki/Rig_(algebra) -- Mike Stay staym(a)clear.net.nz http://www.xaim.com/staym

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Re: [math-fun] Two problems related to recent topics
by asimovd＠aol.com 03 Jul '03

03 Jul '03

1. Suppose a cylindrical can C is inscribed in a unit cube so its axis lies along the main diagonal of the cube. Its radius can be anywhere between 0 (vol(C) = 0) and the radius of the circle inscribed in the hexagonal cross-section of the cube (vol(C) = 0). QUESTION: What radius maximizes the can's volume? 2. Let Z+ be the positive integers and let a > 0 be an irrational real. Define F: Z+ --> Z+ via F(n) = #( [n,n+1) int { K + L*a : K,L are in Z+} ). QUESTION: Find an asymptotic expression for F(n) as n --> oo. --Dan A.

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Re: [math-fun] triple star
by asimovd＠aol.com 03 Jul '03

03 Jul '03

Bill G. wrote: << Wouter suggested http://www.myphysicslab.com/collision.html . That's very nice, but I meant one 3D brick tumbling without collisions in 0g. >> I wrote software (for SGI machines, using GL) to do exactly this in 1987. I might be able to dig it up and it *might* be compatible with OpenGL, but I don't know what the chances are. (Maybe someone else on the list still has a copy of it handy?) Learned a lot about the Jacobi elliptic functions sn, cn, dn in the process, most of which I've forgotten by now. Most interesting thing was to be able to graphically duplicate with real-time computation what happens when one tries to flip a generic rectangular solid through the air 180 degrees about its middle axis. If you aren't familiar with this, I won't spoil your surprise. Go get a typical book, (WLOG size A x B x C with A < B < C), keep it from opening with a strong rubber band, and try to flip it through the air 180 degrees about its B axis. --Dan A.

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Re: [math-fun] NY State math exam flapment
by Dan Hoey 03 Jul '03

03 Jul '03

Henry Baker <hbaker1(a)pipeline.com> > From what I can tell, these tests are already doing what they are > intended to do: determine who can function in our society. In that case, the "Math A Regents exam" is not a mathematics test, so your remark is misdirected. This is a list about mathematics, not cultural integration tests. Not that your departure from the topic is comparable to the reek of gas about "flaming liberals" we got from the flaming ... whatever ... earlier today. That one belongs on a bus--they take any kind of rant you like there. Meanwhile, I'm intrigued by the straw problem. So far, the best I've done is to express the length as sqrt(tt+uu+vv) where (t,u,v) satisfy equations (x-t)^2 (tt+uu+vv) = 4 r^2 (uu+vv) (y-u)^2 (tt+uu+vv) = 4 r^2 (tt+vv) (z-v)^2 (tt+uu+vv) = 4 r^2 (tt+uu) in the box dimensions (x,y,z) and the straw radius r. Adding them up, I'm startled (possibly not for the first time) that any way you lean a disc of radius r on the floor against two walls, the center will be sqrt(2)r from the corner. Is there an easy geometrical proof? But I'm pretty much flummoxed about solving the whole system. Maybe I should have gone to high school in New York. Dan Hoey(a)AIC.NRL.Navy.Mil

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Re: [math-fun] Re: Z*(SqrtN)
by Dan Hoey 03 Jul '03

03 Jul '03

Marc LeBrun <mlb(a)fxpt.com> wrote > > I take it "Z*" is the nonnegative integers. > Yes. I've never much liked that notation either. Is there a better > one? I've also considered using Z0, N0 and even U (for unsigned) but Z* > seems pretty prevalent, despite its opacity. I like Z^(>=) (i.e. Z superscript greater-than-or-equal). Then you can use things like Z^(>1) for {2,3,...}. Let a countable number of flowers bloom! > >>=Henry Baker > >> "algebraic natural numbers" or "natural algebraic numbers" ?? > > Cute name, but I'd rather use that for the positive elements of > > Z(SqrtD) (or the nonnegative ones, in France and set theory). > The "non-negative algebraic integers" might be construed to include > the undesired Sqrt2-1, which is positive but has a negative > component. Yes, that's what I meant. I mostly like this definition of natural because it leaves useful definitions for "supernatural" and "subnatural". Of course, it's mostly because I get a kick out of naming "supernatural numbers". I imagine Knuth got a similar kick out of naming Conway's "surreal numbers". > > Then the elements of Z(SqrtD) with both components positive could > > be called "supernatural", and those with at least one component > > positive could be called "subnatural". > Isn't "natural" supposed to connote a strictly positive integer? In Germany and algebra, yes. In France and number theory, zero is natural. That's of course a gross (and possibly geographically incorrect) simplification of the zerophilia question, but basically you can't count on a particular meaning of "natural" unless you define it. For most natural purposes, it doesn't really matter--you can get a peano with or without the "Z" key, and you just have to transpose music written for the other kind. > I would think "natural algebraic numbers" would refer to only those > numbers with all components positive integers. I guess if you like N = Z^(>), that's a good way of thinking. In that case, I'd call Z^(>=) the "znatural numbers". Just be careful saying it, because people start slapping for mosquitos. > Assuming that, I guess it'd make (some kind of) sense to call the > superset got by adjoining the ("unnatural"?<;-) elements with zero > components the "supernatural algebraic numbers". I still like "super-" = both components, "sub-" = either component definition. Then if you want both components znatural, you're talking about the superznatural part of Z(sqrt2). Dan Hoey(a)AIC.NRL.Navy.Mil

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[math-fun] Mathematics testing
by Eugene Salamin 03 Jul '03

03 Jul '03

I have two suggestions for improving the art of mathematics testing. 1. Cultural bias. Formulate the questions culture-independently. For example "What is the length of the diagonal of a 7 by 11 by 13 inch rectangular solid?" 2. Incorrectly stated problems, incorrect official answer, ambiguous interpretations. Post the exam questions and multiple choice answers on the internet, so anybody can criticize them. Build up a huge inventory of questions, all publicly available. Badly worded or ambiguous questions are fixed or removed. Questions are at all levels of difficulty. Exam administrators choose exam questions from this validated library. Students can access the library as part of their exam preparation. Will students memorize answers instead of learning the concepts? Perhaps, but by the time the library reaches several million questions, even the laziest student will be motivated to learn algorithms and concepts. Design the process so that teaching mathematics and teaching to the exam become coincident. __________________________________ Do you Yahoo!? SBC Yahoo! DSL - Now only $29.95 per month! http://sbc.yahoo.com

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Re: [math-fun] The natural numbers start with 2
by Marc LeBrun 03 Jul '03

03 Jul '03

>=Willem Nienhuys <wsadjw(a)rw7.urc.tue.nl> > The natural numbers start with 2 (TWO). Didn't the ancient Greeks seriously take something like this position? Which reminds me to ask: How did those guys actually perform divisions? Was there some long-division like algorithm used in antiquity, or did they do something else? Of course a classical author might just state the result without showing the work, but there must be some leftover scraps of papyrus or something lying around...

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Re: [math-fun] NY State math exam flap
by Henry Baker 03 Jul '03

03 Jul '03

I haven't had time to look at the test itself, so I will rely on your thoughts. Word problems are _always_ somewhat ambiguous; I think that that is the point of them. They test the student's ability to use common sense and abstract the important elements. That's why the easiest way to confuse student in word problems is to give them extraneous information. They keep trying to incorporate all the information, and all of the information isn't necessary. Are you telling me that in this era where _everyone_ practises for tests, using Kaplan, internet, whatever, that students aren't prepared for these kinds of tests? I keep hearing stories about teachers teaching _only_ for the tests. One would expect that under these conditions, the students would do well on the tests, but still not know any math. But we have an even worse situation -- they don't do well on the tests, and they don't know math, either. Perhaps students did better (as when we were in high school) when they didn't actually "study" for the tests. At 08:22 AM 7/3/03 -0400, William Thurston wrote: >On Thursday, July 3, 2003, at 12:35 AM, Henry Baker wrote: > >I looked at the exam on the web. There were many questions that to me >seemed poorly designed. This particular question as you paraphrased it >may be something that you think most people *should* be able to do, but >in reality I think few high school graduates actually master it, since, >in particular, solid geometry is generally not taught. The actual >question was about a juice container something like 2" x 4" x 5", and >it asked how long a straw would fit diagonally, to the nearest tenth of >an inch. A straw is not the same as a line, and the thickness of the >straw (not specified, of course, and *intended* to be disregarded) >would actually be likely to change the answer. People who've been >taught a particular mechanical formula for this particular question can >do it, but otherwise, it's likely to be a time-consuming and futile >exercise, compounded by trying to calculate square roots presumably >without a calculator.

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[math-fun] NY State math exam flap
by Henry Baker 03 Jul '03

03 Jul '03

Does anyone know any more about the problems referred to below with the NY State Math "Regents exam"? In the NYTimes there was an example of one of the problems on the test, which simply asked what the diagonal distance of a rectangular (3D) box is in terms of its sides -- not an especially difficult question for a junior or a senior in high school who has even a smattering of math and/or physics. I guess this brings up the age-old question of "how much math _should_ the average person who graduates high school should know?" A lot of students think that math is really out of touch with reality. E.g., in today's world of ubiquitous calculators, why waste time on teaching long division? A more relevant question (which my gardener/landscaper just failed) is "what sprinkler pattern covers the grass best?" (Answer is _not_ a cartesian grid.) So clearly high schools aren't teaching even highly relevant math. The diagonal distance of a rectangular box may not be the most relevant question, but I just met a decorator who flunked the 2D version. Shelves in kitchens are priced by the "linear foot"; how many feet of shelves do you have if the shelves are at an angle? (Dunno.) We now have a nation full of math idiots! I have a solution: we give a test to everyone who wants to immigrate to the U.S. If you pass the test, you get in; if you don't, you don't. Since the U.S. birth rate is falling (as it is in every "first world" country), at least we can get an educated population through immigration. ----- http://www.nytimes.com/2003/07/02/nyregion/02REGE.html?pagewanted=print&pos… ------ July 2, 2003 Math Failures Are Raising Concerns About Curriculum By KAREN W. ARENSON Everyone knows now that the June 17 Math A Regents exam was problematic, with almost two out of three test-takers failing. The question is why. Richard P. Mills, the state education commissioner, nullified the test results last week for juniors and seniors, and he and the Regents will soon appoint a panel of experts to figure out what went wrong. In the meantime, parents, teachers and others have their own theories. Some of their explanations have to do with the mechanics of the test: poorly worded questions, lengthy questions that required students to be good readers, multiple-choice questions that had more than one valid answer. Some educators say they were the sort of problems that could have been caught by closer scrutiny of the test before it went out. "If someone had paid me $100 to proof this test, I could have picked out the problems that would create problems," said Daniel Jaye, chairman of the math department at Stuyvesant High School in Manhattan. But some explanations touch on deeper issues, including whether the Math A curriculum is too broad, how much harder it is for students to solve problems than to manipulate equations, and whether unqualified teachers are even less likely to succeed in preparing students than they were with the old math curriculum. When the Math A exam was introduced in 1999, the State Education Department proudly declared that it was harder than Math 1, the old first-level test. There would be more word problems, and students would need a deeper understanding of math and math concepts to do well. Instead of covering a year's worth of material, as most Regents exams do, Math A was meant to be learned over three semesters. Although some students complete the work in a year, some schools have added extra instruction periods and tutoring sessions, and have stretched the course out over four semesters. For a while, the state offered both the old Math 1 exam and the new Math A test. Some schools moved their students quickly to Math A, while others dragged out the transition. The Math 1 test was given for the last time in January 2002. Alan Ray, a spokesman for the Education Department, said that the scores on Math A had gradually improved. In June 2002, 86,000 students took the test, and the passing rate was 61 percent. The department does not yet know how many students took the test last month, but a preliminary survey suggests that only 37 percent of the students passed. The shift from rote learning to a greater emphasis on mastery of concepts is welcomed by some college professors in math and science, who have been trying to accomplish the same shift. They say that although mastery of some facts is critical, students who focus on memorization may do well in a course but remember little of it six months later. "Many of the students are coming from an environment where the tests are pretty much about facts, not about how to puzzle something out, or whether the data you have been given makes sense," said Robert B. Suter, a biologist at Vassar College. "It takes students coming in a while to learn that they are in a somewhat different ballgame." But high school math teachers in New York say that the shift in the state math curriculum has not been smooth, and that schools, teachers and students could not keep up with the changes. "There has to be a balance between skills and concepts," said Mr. Jaye at Stuyvesant. "But the weight shifted a little too quickly for us to keep equilibrium here, and collectively we capsized. "I'm not a champion of the old Regents, which was cookie cutter all skills. If you studied the three previous exams, there would be no surprises. But in the effort not to make Math A cookie cutter, the kids got flamed." Part of the problem, he and others said, is that the curriculum expanded and expanded, and nothing was taken away. Many teachers felt that they were racing through material without giving students the time they needed to master it. "One of the big problems is that the curriculum is so broad that you cannot go very deep," said Bob Hazen, president of the Association of Mathematics Teachers of New York State. "It used to be with the old Math 1, 2 and 3 tests that you could just bone up on old tests and pass them and not have a firm grasp of the concepts," Mr. Hazen said. "But now you are looking at a standards-based assessment, and it is very difficult to look at the past tests and know what will be on the next test." Some educators say teaching students to be problem solvers takes more skill on the part of teachers, a challenge when there is a shortage of qualified math teachers. "Teachers are not really prepared to prepare kids for this test properly," said Alfred S. Posamentier, dean of the School of Education, City College of New York, and the author of books on problem solving. "There is very little training for teachers in problem solving; it's assumed they will get it along the way." Commissioner Mills said he stood by the Math A curriculum. "There are a great many people who say we should just teach students how to balance a checkbook," he said. "But that is a level of math that is learned by the fourth grade. Are we ready to say that the most minimal level is all students need?" "The Regents made the fundamental policy choice here, and I support it," he added. "The standards represent a level of knowledge and skill that's essential for every child." Copyright 2003 The New York Times Company | Home | Privacy Policy | Search | Corrections | Help | Back to Top

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Re: [math-fun] the negative-third digit of pi, etc.
by Henry Baker 03 Jul '03

03 Jul '03

I actually considered this in the case of reversing a Newton's iteration a few years ago. Consider, for example, reversing a Newton's iteration for the sqrt of a number. Unfortunately, the commutativity of "+" loses 1 bit of information per iteration, _even when we preserve infinite precision -- i.e., by starting with a rational number and keeping rational numbers_ (e.g., do this with Macsyma/Maxima/Maple/etc.). Thus, (x+N/x)/2 = (y+N/y)/2, where y=N/x, because N/y=N/(N/x)=x. We will ignore this issue by making some sort of consistent choice. We now get chaotic behavior -- even the smallest change in the initial conditions produces very large changes in the succeeding computation. But this is the whole point of the Newton iteration -- to eventually ignore the initial "guess" and converge on the answer! Thus, I concluded that convergence == reversed chaos! At 10:28 AM 7/3/03 -0500, James Propp wrote: >Most of us know (and if you don't already, you will by the end of this >sentence!) that there are models of computation that are universal (they >can simulate a Turing machine running an arbitrary program with arbitrary >input) and also reversible (each state of the computation determines its >predecessor). > >My question is, what happens if one uses reversibility to run a >mathematically interesting computation into negative time? > >E.g., suppose one has a reversible machine that, when run forward, computes >the sequence of primes, or the digits of pi, or something like that. Is >there anything interesting that happens when you run the machine backwards? > >To make sense of this question, one needs to be precise about what it means >to say that a machine computes something. Often there is some non-trivial >representation scheme; e.g., in (non-reversible) Life-simulations, a sequence >might be represented by the spacings of some gliders. > >It could be that if you run a computation backwards from zero, properties >of the representation that hold at positive times cease to hold, so that >there's no canonical way to extend the _representation_ of the computation >as a sequence of digits or whatever. > >My guess is that for most interesting mathematical questions with answers >indexed by the natural numbers, and for most reversible computation schemes >for computing those answers, you don't get anything interesting or canonical >by running the computation backwards. But it would still be interesting to >see what happens if one tries. > >I should mention that one slight exception to my overall pessimism: namely, >there are reversible (albeit non-universal) arithmetic models of computation >which when run forward can compute good rational approximations to a >quadratic irrational and when run in reverse compute good approximations >to its algebraic conjugate. But this is a very special situation. And >indeed, if one uses a similar model of computation to try to compute rational >approximations to e, one finds that if one runs the program in reverse, it >goes into an infinite loop when it tries to divide 1 by 0. > >Jim Propp

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