July 2012 - mathfun@mailman.xmission.com

[math-fun] why can a line of slope 10^n determine n decimal digits of π?
by Gary Antonick 19 Jun '14

19 Jun '14

Just noticed this yesterday and can't figure out why it works. Draw a circle with radius 1 around 0,0. Draw a line with slope -10 that hits the circle at 1,0. This line will also hit the circle again and make a little arc. The length of the arc will be about 1/31 the perimeter of the circle. Now try again with a line of slope -100. The line will create an arc 1/314 of the circle. Starting to look a bit like pi. Am wondering why this is. In case I made a mistake here's what I'm trying to describe.. if the circle were a clock the first line would cut an arc from 15 minutes (3:00) to almost the 13 minute mark.

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[math-fun] sqrt(pi)
by Bill Gosper 24 Mar '13

24 Mar '13

Pairwise combining the terms of A&S 6.5.31 (upper Gamma CF), then adding last week's lower gamma CF: In[185]:= Equal[Gamma[ z], ((y^z*((1/(cfk[n*(z - n), -z + y + 2*n + 1, List[n, Infinity]] - z + y + 1)) + (1/(z + cfk[n*y, z - y + n, List[n, Infinity]] - y))))/(E^y))] z 1 Gamma[z] == (y (------------------------------------------- + -y + z + cfk[n y, n - y + z, {n, Infinity}] 1 y -----------------------------------------------------------)) / E 1 + y - z + cfk[n (-n + z), 1 + 2 n + y - z, {n, Infinity}] where cfk:=ContinuedFractionK and y>0 can be chosen to optimize convergence. Specializing, In[186]:= % /. z -> 1/2 /. y -> 1 Out[186]= 1 Sqrt[Pi] == (-------------------------------------- + 1 1 -(-) + cfk[n, -(-) + n, {n, Infinity}] 2 2 1 ------------------------------------------) / E 3 1 3 - + cfk[(- - n) n, - + 2 n, {n, Infinity}] 2 2 2 In[192]:= N[List @@ %186 /. cfk[num_, den_, _] :> ContinuedFractionK[num, den, {n, 9}]] Out[192]= {1.77245, 1.77244} This is the first non-asymptotic sum or CF I've seen for sqrt(pi). But the sum behind the lowergamma cf has been there all along, so I must not have been paying attention. --rwg

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[math-fun] peculiar 2x2 products
by Bill Gosper 15 Aug '12

15 Aug '12

We're used to evaluating an infinite continued fraction as a limit of ratios of elements of a running product of 2x2 matrices, while the actual elements of the product blow up. E.g., for 4 arctan(1) In[1153]:= FoldList[Dot, {{0, 4}, {1, 0}}, Table[{{2*k - 1, k^2}, {1, 0}}, {k, 9}]] Out[1153]= {{{0, 4}, {1, 0}}, {{4, 0}, {1, 1}}, {{12, 16}, {4, 4}}, {{76, 108}, {24, 36}}, {{640, 1216}, {204, 384}}, {{6976, 16000}, {2220, 5100}}, {{92736, 251136}, {29520, 79920}}, {{1456704, 4544064}, {463680, 1446480}}, {{26394624, 93229056}, {8401680, 29675520}}, {{541937664, 2137964544}, {172504080, 680536080}}} In[1154]:= N[Divide @@ # & /@ Rest[%]] Out[1154]= {{4., 0.}, {3., 4.}, {3.16667, 3.}, {3.13725, 3.16667}, {3.14234, 3.13725}, {3.14146, 3.14234}, {3.14161, 3.14146}, {3.14159, 3.14161}, {3.14159, 3.14159}} But here's an infinite matrix product where the elements converge individually: Product[MatrixForm[{{(3*a^2 - 3*a*j + j^2)/(j*(-1 - 3*a + b + j)), a^3/(j*(-1 - 3*a + b + j))}, {1/j, 1}}], {j, b, ∞}] == MatrixForm[{{"?", (a^3*HypergeometricPFQ[{1 + a, 1 + a, 1 + a}, {1 + b, 1 + b}, 1])/b^2}, {"?", HypergeometricPFQ[{a, a, a}, {b, b}, 1]}}] (Product means matrix product, not Mathematica's product.) The ?s converge, too, but I don't know to what. Note that the determinant ≠ 1, but approaches 1 in a very particular way. It's not obvious to me how to scale a matrix to make its running product converge elementwise. I actually have an elementwise convergent 2x2 product for the general 3F2[1], but idiot Mathematica 8.04 takes (literally?) forever to convert a simple sum of Pochhammers to a 3F2. --rwg

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[math-fun] Is algebra necessary?
by Warren Smith 06 Aug '12

06 Aug '12

For what it is worth, I think it would be absolutely absurd to allow getting a CS college degree without knowing calculus. I also have a hard time accepting the idea that general people do not need to know and should not be taught algebra. But the best argument for that stance might be (which is not mentioned at all in the newspaper of course) the advent of software like mathematica. I think that technology like that PERHAPS could be used to completely change traditional math education and both speed it up and make it better. Optimally, you'd learn the principles and ideas while skipping the drudgery and errors. I'm not sure if that goal is feasible, but I am fairly sure that the educators and software writers haven't yet put in a high quality attempt to accomplish it. There have been low-quality attempts which I never saw convincing evidence worked better than 1950s style teaching and may work worse. One thing I'd like to see in education is controlled experiments, like educate class #K with method #K. This would be easy to do in large universities. But I've never seen any ever do it.

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Re: [math-fun] We don't need no stinking algebra
by Dave Dyer 02 Aug '12

02 Aug '12

> >The responses were all "duh! You press the 'sin' key! ...So since we >can do that, Yes, but suppose you were shipwrecked on a desert island with nothing but a sextant, a digital watch, a slide rule, and ONE bottle you could cast into the sea with your latitude and longitude written on it.

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[math-fun] name for 1/8 of sphere
by James Buddenhagen 01 Aug '12

01 Aug '12

A friend who is a sculptor asked me if there is a name for the shape of the 8 pieces a solid sphere would fall into if you cut it with 3 mutually perpendicular planes through the center. I couldn't think of any but thought I would ask here. And more generally, is there a name for a 'spherical wedge' bounded by a spherical triangle and the planar faces from the center containing the triangle's edges? Thanks for any help on this. Jim

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[math-fun] Bill and Richard make (another) appearance in TAOCP
by quad 31 Jul '12

31 Jul '12

Knuth recently just put out a draft of his section on satisfiability (section 7.2.2.2, available here: www-cs-faculty.stanford.edu/~knuth/fasc6a.ps.gz) and there they are, Gosper and Schroeppel referenced in the answers to the exercises, page 60. Here's the excerpt (as an image): [SPOILER, answer to exercise 9d] http://i.imgur.com/IXSIH.png [/SPOILER] Three cheers for HAKMEM! -Robert

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[math-fun] We don't need no stinking algebra
by Warren Smith 31 Jul '12

31 Jul '12

> The intersection of people who struggle with basic algebra and the > people who use Mathematica has a cardinality equal to the infimum > of positive reals. :P > Sincerely, > Adam P. Goucher --yes, but that is kind of my point. Does it have to be that way? Maybe they'd struggle less with appropriate computer help? I don't know. Partly, you cannot use such tools unless you already know what you are doing, so it is somewhat pointless... but those tools DO know what they are doing, at least about basic algebra, so it seems in principle, it ought to be possible for Joe Schmoe to do algebra even if fairly clueless, if has help from software that knows how. For example, Joe Schmoe can do long division with calculator aid, even though he himself unaided might have no idea how -- a fact the UK education system now agrees with. One time I was teaching calculus I wanted to tell them about Maclaurin series and was motivating them by asking "how does your calculator compute sin(50 degrees)?" The responses were all "duh! You press the 'sin' key! ...So since we can do that, why do we care?" Which was a quite defensible reaction from them all, even though not what I wanted to hear :( -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] NYTimes: Is Algebra Necessary?
by rcs＠xmission.com 31 Jul '12

31 Jul '12

I've been meaning to mention the article "What Is Mathematics For?" by Underwood Dudley. The title should be "What Is Mathematics Education For?" www.ams.org/notices/201005/rtx100500608p.pdf A short version of his case is "Is Mathematics Necessary?" at http://www.public.iastate.edu/~aleand/dudley.html Dudley refutes the claim 'Mathematics is Necessary for Work'. He aims a little higher than Algebra, but presents a good set of examples. The finale is a plea for 'Math as Art', asking readers to look beyond the pedestrian goal of preparing students for a job. (A tough sell during the Great Recession.) He does mention the justifications 'Math teaches logical thinking', and the general 'we want smart people here, and math is a good filter'. The issue came up in the Arizona CS department c. 1995, when the curriculum committee was debating whether to keep the Calculus requirement for the CS degree. (CS is a curious meld: programming skill and algorithm analysis, blended with the advanced math of Relative Computability.) Rather than reprise the arguments pro & con, I'd like to pose a puzzle: Why do so many people find math too hard? I think the toughest memorization task is learning the multiplication table, and the most complex widely taught algorithm is long division. Both of these are now 'calculator' jobs, which ought to make easier the problem of solving 2x = 5. Somehow, it hasn't. What's the difficulty? Rich ----- Quoting Henry Baker <hbaker1(a)pipeline.com>: > FYI -- I'm not fond of the way algebra is taught in schools, but I > think it would be insane not to teach it at all. The comments on > this article are interesting. > > http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html > > Opinion > > Is Algebra Necessary? > > Adam Hayes > By ANDREW HACKER > Published: July 28, 2012 > > A TYPICAL American school day finds some six million high school > students and two million college freshmen struggling with algebra. > In both high school and college, all too many students are expected > to fail. Why do we subject American students to this ordeal? Ive > found myself moving toward the strong view that we shouldnt.

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Re: [math-fun] Is algebra necessary?
by Henry Baker 31 Jul '12

31 Jul '12

Before trying to teach algebra with Mathematica or Maxima, I'd first start with a spreadsheet like Excel. There's a heck of a lot of math that can be taught with a spreadsheet, and these languages also have plotting/graphing. Also, being able to use a spreadsheet is good even for art students who want to keep track of their artwork, and for lawyers who want to keep track of their clients. At 10:03 AM 7/30/2012, Warren Smith wrote: >For what it is worth, I think it would be absolutely absurd to allow >getting a CS college degree without knowing calculus. > >I also have a hard time accepting the idea that general people do not >need to know and should not be taught algebra. But the best argument >for that stance might be (which is not mentioned at all in the >newspaper of course) the advent of software like >mathematica. I think that technology like that PERHAPS could be used >to completely change traditional math education and both speed it up >and make it better. Optimally, >you'd learn the principles and ideas while skipping the drudgery and errors. >I'm not sure if that goal is feasible, but I am fairly sure that the >educators and software writers haven't yet put in a high quality >attempt to accomplish it. There have been >low-quality attempts which I never saw convincing evidence worked better than >1950s style teaching and may work worse. One thing I'd like to see in education is >controlled experiments, like educate class #K with method #K. This would >be easy to do in large universities. But I've never seen any ever do it.

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