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  • Joseph Nebus 8:27 pm on Tuesday, 1 September, 2015 Permalink | Reply
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    How August 2015 Treated My Mathematics Blog 


    August was my first full month after the end of the A-to-Z project. How would dropping from the nearly-daily publication in June and early July to a relatively sedate three times a week affect my readership and reader engagement? As ever, the data is mixed.

    According to WordPress’s counters, the number of page views rose to 909, up from July’s 863 and down from busy June’s 1,051. I like the upward trend, especially since the number of unique visitors rose to 506. That’s a record high. It’s up appreciably from July’s 415 and June’s 367. However, I have to suspect the numbers. On the 20th of August the blog came under the attention of what looks like some kind of content aggregator or advertising site. That sent me 90 clicks on the 20th, and while that’s flattering I have my doubts anybody was actually reading anything.

    The number of likes I received over the month dropped to 296. That’s down from July’s 381 and June’s 518. That sounds like a calamitous drop until you remember that in June I posted 28 things, July 24, and August a mere 14. On a per-post basis that’s slightly over 21 likes per posting in August, up from about 16 in July and 18.5 in June. I suspect there’s no meaningful trend here. The number of comments dropped slightly, with only 95 received in August. There were 100 in July and 114 in June. That’s hardly a difference, though, and it’s a very nice-looking comment-per-post trend.

    The two most popular posts in August were reblogged things. Well, at least people liked what was pointed out to them. After that it was mostly comic strips, which I’m comfortable with. I like the range of topics they inspire me to write.

    1. How I Impressed My Wife: Part 1
    2. Original Problem! Expanding Galaxies and Rates of Change
    3. Reading the Comics, August 10, 2015: How People Think Edition
    4. Do You Have To Understand This?
    5. Reading the Comics, August 3, 2015: Things That Make Me Cranky Edition

    The largest number of readers came from the United States, at 496. That’s just about what came here from the United States in July (502). Second-highest was Canada at 45, and third-highest again the United Kingdom, 35. The Philippines came in fourth, with 26 readers. (They were in fifth with 37 readers in July. Rankings are weird things.) Austria gave me 23 page views, and India and Spain sent 22 each. For India that’s a fair jump over July’s 14.

    Single-reader countries in August were quite a list: Argentina, Bangladesh, Belarus, Bulgaria, Colombia, Denmark, Indonesia, Jordan, Kazakhstan, Kenya, Kuwait, Morocco, Nepal, New Zealand, Palestinian Territories, Portugal, Senegal, Slovakia, South Korea, Sudan, Switzerland, Tunisia, and Venezuela. Denmark, Nepal, and Portugal were also single-reader countries in July and nobody’s on a three-month streak there.

    Among the search terms intriguing me the past month were:

    • apollo 13 rewritten checklist
    • origin is the gateway to your entire gaming universe
    • fox trot impossible bobby driving math problem gocomics
    • “the price is right” any number game
    • bringing breakfast cartoons
    • 86164 seconds
    • electro mechanical research sarasota
    • almost everywhere concept

    Of course I understand the “86164 seconds” query, because that ties in to the delightful animated movie Arthur Christmas and the deep existential dread one of its plot points inspires.

    August ends with the blog having received 27,381 total page views, and 9,967 distinct visitors. There’s 517 people listed as WordPress followers. And my comments stand at 2,054. I don’t know who had lucky number 2,038th.

    Finally I should include a reminder to folks who are reading this blog post to read these blog posts. There’s this “Follow Blog via Email” link on the upper right of the page. If you have an RSS reader, https://nebusresearch.wordpress.com/feed/ will give you posts. And my Twitter account is @Nebusj. Thanks for existing and all that.

     
    • Angie Mc 11:44 pm on Wednesday, 2 September, 2015 Permalink | Reply

      How have I missed your twitter account until now? Following :D

      Like

  • Joseph Nebus 8:01 pm on Sunday, 30 August, 2015 Permalink | Reply
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    Reading the Comics, August 29, 2015: Unthemed Edition 


    I can’t think of any particular thematic link through the past week’s mathematical comic strips. This happens sometimes. I’ll make do. They’re all Gocomics.com strips this time around, too, so I haven’t included the strips. The URLs ought to be reasonably stable.

    J C Duffy’s Lug Nuts (August 23) is a cute illustration of the first, second, third, and fourth dimensions. The wall-of-text might be a bit off-putting, especially the last panel. It’s worth the reading. Indeed, you almost don’t need the cartoon if you read the text.

    Tom Toles’s Randolph Itch, 2 am (August 24) is an explanation of pie charts. This might be the best stilly joke of the week. I may just be an easy touch for a pie-in-the-face.

    Charlie Podrebarac’s Cow Town (August 26) is about the first day of mathematics camp. It’s also every graduate students’ thesis defense anxiety dream. The zero with a slash through it popping out of Jim Smith’s mouth is known as the null sign. That comes to us from set theory, where it describes “a set that has no elements”. Null sets have many interesting properties considering they haven’t got any things. And that’s important for set theory. The symbol was introduced to mathematics in 1939 by Nicholas Bourbaki, the renowned mathematician who never existed. He was important to the course of 20th century mathematics.

    Eric the Circle (August 26), this one by ‘Arys’, is a Venn diagram joke. It makes me realize the Eric the Circle project does less with Venn diagrams than I expected.

    John Graziano’s Ripley’s Believe It Or Not (August 26) talks of a Akira Haraguchi. If we believe this, then, in 2006 he recited 111,700 digits of pi from memory. It’s an impressive stunt and one that makes me wonder who did the checking that he got them all right. The fact-checkers never get their names in Graziano’s Ripley’s.

    Mark Parisi’s Off The Mark (August 27, rerun from 1987) mentions Monty Hall. This is worth mentioning in these parts mostly as a matter of courtesy. The Monty Hall Problem is a fine and imagination-catching probability question. It represents a scenario that never happened on the game show Let’s Make A Deal, though.

    Jeff Stahler’s Moderately Confused (August 28) is a word problem joke. I do wonder if the presence of battery percentage indicators on electronic devices has helped people get a better feeling for percentages. I suppose only vaguely. The devices can be too strangely nonlinear to relate percentages of charge to anything like device lifespan. I’m thinking here of my cell phone, which will sit in my messenger bag for three weeks dropping slowly from 100% to 50%, and then die for want of electrons after thirty minutes of talking with my father. I imagine you have similar experiences, not necessarily with my father.

    Thom Bluemel’s Birdbrains (August 29) is a caveman-mathematics joke. This one’s based on calendars, which have always been mathematical puzzles.

     
  • Joseph Nebus 7:20 pm on Friday, 28 August, 2015 Permalink | Reply
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    Do You Have To Understand This? 


    At least around here school is starting up again and that’s got me thinking about learning mathematics. Particularly, it’s got me on the question: what should you do if you get stuck?

    You will get stuck. Much of mathematics is learning a series series of arguments. They won’t all make sense, at least not at first. The arguments are almost certainly correct. If you’re reading something from a textbook, especially a textbook with a name like “Introductory” and that’s got into its seventh edition, the arguments can be counted on. (On the cutting edge of new mathematical discovery arguments might yet be uncertain.) But just because the arguments are right doesn’t mean you’ll see why they’re right, or even how they work at all.

    So is it all right, if you’re stuck on a point, to just accept that this is something you don’t get, and move on, maybe coming back later?

    Some will say no. Charles Dodgson — Lewis Carroll — took a rather hard line on this, insisting that one must study the argument until it makes sense. There are good reasons for this attitude. One is that while mathematics is made up of lots of arguments, it’s also made up of lots of very similar arguments. If you don’t understand the proof for (say) Green’s Theorem, it’s rather likely you won’t understand Stokes’s Theorem. And that’s coming in a couple of pages. Nor will you get a number of other theorems built on similar setups and using similar arguments. If you want to progress you have to get this.

    Another strong argument is that much of mathematics is cumulative. Green’s Theorem is used as a building block to many other theorems. If you haven’t got an understanding of why that theorem works, then you probably also don’t have a clear idea why its follow-up theorems work. Before long the entire chapter is an indistinct mass of the not-quite-understood.

    I’m less hard-line about this. I’m sure that shocks everyone who has never heard me express an opinion on anything, ever. But I have to judge the way I learn stuff to be the best possible way to learn stuff. And that includes, after a certain while of beating my head against the wall, moving on and coming back around later.

    Why do I think that’s justified? Well, for one, because I’m not in school anymore. What mathematics I learn is because I find it beautiful or fun, and if I’m making myself miserable then I’m missing the point. This is a good attitude when all mathematics is recreational. It’s not so applicable when the exam is Monday, 9:50 am.

    But sometimes it’s easier to understand something when you have experience using it. A simple statement of Green’s Theorem can make it sound too intimidating to be useful. When you see it in use, the “why” and “how” can be clearer. The motivation for the theorem can be compelling. The slightly grim joke we shared as majors was that we never really understood a course until we took its successor. This had dire implications for understanding what we would take senior year.

    What about the cumulative nature of mathematical knowledge? That’s so and it’s not disputable. But it seems to me possible to accept “this statement is true, even if I’m not quite sure why” on the way to something that requires it. We always have to depend on things that are true that we can’t quite justify. I don’t even mean the axioms or the assumptions going into a theorem. I’m not sure how to characterize the kind of thing I mean.

    I can give examples, though. When I was learning simple harmonic motion, the study of pendulums, I was hung up on a particular point. In describing how the pendulum swings, there’s a point where we substitute the sine of the angle of the pendulum for the measure of the angle of the pendulum. If the angle is small enough these numbers are just about the same. But … why? What justifies going from the exact sine of the angle to the approximation of the angle? Why then and not somewhere else? How do you know to do it there and not somewhere else?

    I couldn’t get satisfying answers as a student. If I had refused to move on until I understood the process? Well, I might have earlier had an understanding that these sorts of approximations defy rigor. They’re about judgement, when to approximate and when to not. And they come from experience. You learn that approximating this will give you a solvable interesting problem. But approximating that leaves you too simple a problem to be worth studying. But I would have been quite delayed in understanding simple harmonic motion, which is at least as important. Maybe more important if you’re studying physics problems. There have to be priorities.

    Is that right, though? I did get to what I thought was more important at the time. But the making of approximations is important, and I didn’t really learn it then. I’d accepted that we would do this and move on, and I did fill in that gap later. But it is so easy to never get back to the gap.

    There’s hope if you’re studying something well-developed. By “well-developed” I mean something like “there are several good textbooks someone teaching this might choose from”. If a subject gets several good textbooks it usually has several independent proofs of anything interesting. If you’re stuck on one point, you usually can find it explained by a different chain of reasoning.

    Sometimes even just a different author will help. I survived Introduction to Real Analysis (the study of why calculus works) by accepting that I just didn’t speak the textbook’s language. I borrowed an intro to real analysis textbook that was written in French. I don’t speak or really read French, though I had a couple years of it in middle and high school. But the straightforward grammar of mathematical French, and the common vocabulary, meant I was able to work through at least the harder things to understand. Of course, the difference might have been that I had to slowly consider every sentence to turn it from French text to English reading.

    Probably there can’t be a universally right answer. We learn by different methods, for different goals, at different times. Whether it’s all right to skip the difficult part and come back later will depend. But I’d like to know what other people think, and more, what they do.

     
    • Angie Mc 9:11 pm on Friday, 28 August, 2015 Permalink | Reply

      I don’t have an answer, Joseph, but you have given me a great post to share with my teen. He is an atypical non-linear learner, but learn he does. Personally, I perceive myself as a lifelong learner so I lean toward moving on from difficult material with the hope of many opportunities to revisit it later, perhaps with more experience (wisdom, patience, etc.)

      Liked by 1 person

      • Joseph Nebus 8:10 pm on Sunday, 30 August, 2015 Permalink | Reply

        I hope he’s able to draw something useful. You are probably right that there’s a difference between lifelong learners — well, let’s call them recreational learners — and people who learn because they want to get a specific skill or block of knowledge. A recreational learner probably has an easier time of letting a difficult point go and coming back to it later, not least because there isn’t a deadline demanding knowledge of this particular thing by this particular time.

        Liked by 1 person

        • Angie Mc 11:24 pm on Monday, 31 August, 2015 Permalink | Reply

          Exactly. One challenge younger students can face is when deadlines come before developmental readiness and/or interest. My goal is to keep my kids in the learning game, to see themselves as learners, until hopefully they get past all the deadlines and can have space to creatively tackle topics that were more difficult when they were younger. Hope your week is off to a great start, Joseph!

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          • Joseph Nebus 5:10 pm on Wednesday, 2 September, 2015 Permalink | Reply

            Thank you; I hope yours is turning out well. I’ve been away from teaching for a couple of years so have been getting rusty in working out how to keep students interested and somewhere near deadline.

            Liked by 1 person

    • Michelle H 10:09 pm on Friday, 28 August, 2015 Permalink | Reply

      When I think back to learning literary theory, in particular deconstruction back in the day when Jaques Derrida was still alive, there was a need for patience, as well as trust that it would make sense one day. Everything was cutting edge and so few people really understand this highly abstract philosophy, so it took time to absorb even the basic structures. To answer your question, I agree that learning requires a lot of suspension. In a couple weeks I start university again, this time studying mathematics, and it helps to remember that I already know my learning process, and it is not so different from the way you’ve described yours. (One less thing to sweat about.)

      Liked by 1 person

      • Joseph Nebus 8:15 pm on Sunday, 30 August, 2015 Permalink | Reply

        I hadn’t thought about the parallels in other fields but certainly a similar process has to apply.

        New or cutting-edge material has to be more difficult to learn, it seems to me. For one even the founding terms have to be more in flux. What the core concepts are, and what their meanings should be, requires experience and practice and a new field isn’t going to have that to draw on. And there’s less time for the development of good introductory exposition, and for alternate lines of presentation, in something new.

        I confess I forget the details now but I remember in differential equations hearing of a cornerstone paper from about 1950 that, allegedly, no one had ever understood from reading it. What knowledge people had of it came from people who’d had it explained by the original author, who went on to explain to their students with only passing reference to the actual paper, and so on. I’m dubious of any claims about an unreadable paper, but the idea of an important paper supported by an oral tradition is the sort of charming thing people don’t realize mathematicians do.

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    • mathtuition88 5:40 am on Saturday, 29 August, 2015 Permalink | Reply

      I think “not understanding the math course until you take its successor” is very true.
      Von Neumann said something like, “One never really understands mathematics, one just gets used to it”.
      Good article!

      Like

      • Joseph Nebus 8:16 pm on Sunday, 30 August, 2015 Permalink | Reply

        Was that mathematics? I’d thought it was quantum mechanics, although there is a point where the difference doesn’t matter. The quote and the sentiment are meaningful either way.

        Liked by 1 person

    • howardat58 2:22 am on Sunday, 30 August, 2015 Permalink | Reply

      My advice to anyone embarking on a new topic in math is to read just the text on first, and maybe second, reading. You may get a clue as to what it’s all about that way. Then go back and try figuring out the symbolic stuff.(a bit at a time, and don’t worry about skipping chunks)

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      • Joseph Nebus 8:18 pm on Sunday, 30 August, 2015 Permalink | Reply

        I hadn’t thought of it, but realize that I do tend to read a mathematics text that way. That is, I’ll do a pass where I just read the text and don’t worry about the equations. Later on I do the equations with only glancing reference to the text. At some point I try them together. This is why it’s very slow reading mathematics, I suppose.

        I’m a bit surprised that everyone who’s commented seems to have a similar, or at least compatible, learning style to me. I’m curious whether that’s really widespread, or if I just give off vibes that attract people with similar styles to mine.

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    • elkement 7:37 am on Monday, 31 August, 2015 Permalink | Reply

      I remember learning physics as quite ‘patchy’. You learn about some things as if they were fundamental axioms, only to discover later you could derive them from something else; sometimes you were not so sure after a while ‘what came first’ actually. Then, suddenly, everything fell in place.
      For example, it is hard to pin down for me how and when torque and angular momentum have been explained really the first time. In the first introduction it felt like a parallel world to force / momentum, introduced by using analogues … but not the full-blown derivation from Newton’s laws of course, using Euler’s angles. I think it is also related to the way and order how math and physics are taught. In my physics degree programme, we had two math lectures in the first year – Linear Algebra and Real Analysis (I think I mentioned before that we don’t had any ‘Generad Ed’ or ‘calculus for everybody’ lectures, these were lectures specifically for math and physics ‘majors’ only).
      Theoretically, these were the perfect preparation for dealing with, say, tensors, and we were given a rigid prove of those theorems in vector analysis. But in parallel to the advanced math lectures we had an ‘intro to physics’ lecture using simpler math, and ‘theoretical physics’ started only one year later – and at that time it was hard to relate the physicist’s way of introducing Gauss and Stokes theorem to the much more rigid way you had learned it already. In Real Analysis the focus was on proofs for those aspects that were done more, well, hand-waving, in physics. For example, in ‘theoretical physics’ it was tacitly assumed that functions in the physical world, like distribution of charges, are sort of well-behaved, without kinks and singularities – so you did not have to make such a fuss over functions being continuous or differentiable :-)
      Feynman once said, in relation to his second volume of lectures (electromagnetism) that he would do a ‘mathematical methods for physicists’ lecture, if he would have to do it again, instead of introducing those theorems together with Maxwell’s equations as it is usually done (and obviously also instead of teaching them in a way unrelated to physics).

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      • Joseph Nebus 5:08 pm on Wednesday, 2 September, 2015 Permalink | Reply

        In thinking about it I realize my introduction to physics was also a patchy thing. I originally attributed that to starting out in an advanced program in high school, and so trying to rush physics and calculus into the heads of high school students who probably weren’t really quite up to it. And then Advanced Placement credit meant I was able to skip the college-level introductory courses that might have been a little more harmonious, or at least might have gone back over the basic ground with a little more system.

        Now, Gauss’s theorem I remember getting in (high school) physics as part of the magic of working out electrostatic problems. I don’t remember there being any attempt to link it to what was going on in calculus. I’m not sure my high school calculus course got that far into multivariable integrals either. We did get to it in vector calculus in college, although there again I’m not sure it was ever specifically linked to what was going on physics. There, though, my skipped courses might have made things worse because I think if I’d been taking electromagnetism at the same time as vector calculus the links between the two would have been more obvious. As it was, I had a bunch of vector calculus tools sitting around for a semester (and summer break) without any use.

        Somehow, I never got into a course that gave me enough to do with tensors to really get them straight. They did get a little mention in Classical Mechanics, although the professor wasn’t very interested in the problems they required so we got just enough to get on to what he really liked. (Mind, he did some great work, including an exam problem I thought, then and now, was brilliant: given this completely different pairwise interaction law, derive the equivalent of Kepler’s laws for point masses, and work out whether solid bodies and point masses orbit in different ways.)

        While Linear Algebra did serve my physics courses I don’t think the links were very clear. It’s something I recognize in hindsight, probably because as physics gets more advanced it turns into many more linear algebra or linear algebra-like problems. Real analysis I don’t think served anything in my physics courses, although I realize it was the foundation on which the quantum chemistry course I taught one semester was built. (They needed anybody who could do it, and I was nearby.)

        Liked by 1 person

  • Joseph Nebus 5:00 pm on Tuesday, 25 August, 2015 Permalink | Reply
    Tags: , longitude,   

    Who Was Jonas Moore? 


    I imagine I’m not the only person to have not realized the anniversary of Jonas Moore’s death was upon us again. Granted he’s not in anyone’s short list of figures from mathematical history. The easiest thing to say about him is that he appears to have coined common shorthands for the trigonometric functions: cot for cotangent, that sort of thing. Perhaps nothing exciting, but it’s something that had to be done.

    Moore’s more interesting than that. The Renaissance Mathematicus has a biographic essay. Particularly of interest is that Moore oversaw the building of the Royal Observatory in Greenwich, and paid for the first instruments put into it. And, with Samuel Pepys, he founded the Royal Mathematical School at Christ’s Hospital, to train men in scientific navigation. As such he’s got a place in the story of longitude, and time-keeping, and our understanding of how to measure things.

    That won’t put him onto your short list of important figures in the history of mathematics and science. But it’s interesting anyway.

     
  • Joseph Nebus 6:00 pm on Sunday, 23 August, 2015 Permalink | Reply
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    Reading the Comics, August 22, 2015: Infinite Probabilities Edition 


    It’s summer, so the rate of mathematically-themed comic strips slowed. That’s fine. I had a very busy, distracting couple of weeks and wasn’t able to keep up reading everything quite so faithfully. As it is you’ll notice I’m posting this without having read all my Sunday comics. They can fit in the next edition, surely.

    Carol Lay’s Lay Lines (August 16) is a nicely mind-expanding story about the clash between incredibly unlikely events and an infinitely large universe. This is also my favorite of this week’s strips.

    Infinity and probability interact in weird ways. Infinitely many chances for something to happen, for example, seem to imply that everything that is even remotely possible ought to happen. However … it doesn’t, really. For example, it’s imaginable that one might flip a fair coin infinitely many times and yet never see a streak where it comes up tails more than ten times in a row. Is that improbable? Sure. But impossible? And if it’s not impossible, then mustn’t it happen, given enough chances to? Mathematics and philosophy blend into one another. We see this often in logic. But probability is another of the fields that stands insistently with one foot in mathematics’s question of “what can we say about this model” and one foot in philosophy’s question of “what does it mean for something to be true”.

    Mark Pett’s Lucky Cow (August 16) is about geometry. To study the real world we use straight lines and perfect circles and right angles and regular polygons and such. None of these things happen in the real world. Even things that could be like them, such as the path of a laser, or the area highlighted by a cone of light falling on the wall, are only close to the line or the ellipse or other shape in our ideal. But the abstractions are such very useful things. Well, Clare can rest assured that stacking the meal trays straight would not, in the end, produce a straight line anyway.

    Johnny Hart and Brant Parker’s The Wizard of Id (Classics) (August 16, originally run August 15, 1965) is, as the punch line mentions, a variation on the monkeys-at-typewriters problem. The King is right that, given enough time, even Sir Rodney’s bound to hit a bullseye. But then see the talk about Lay Lines above.

    'Professor, I ran out of round pans, do you mind if these pies are squared?'

    Mike Peters’s Mother Goose and Grimm for the 21st of August, 2015.

    Mike Peters’s Mother Goose and Grimm (August 21) is the oft-made pun about the area of circles. Of course the blackboard with a lot of formulas is used as signifier for “mathematician” or perhaps “really bright person”. The writing on it is basically nonsense, although I’m curious why so many words are written out.

    Mind, it is often very useful when setting out to write out, in plain English, what your variables represent. It helps set out what you think you’re doing, and check back that you’re doing things that are sensible. But, for example, “R” means “Radius” so very often that it’s silly to write that down. More useful would be “R = Radius Of [ something ]”. And then why write out circumference, and diameter, and circle twice on this board? … Besides the fact that what’s on the board is meaningless, that is, and we shouldn’t bother reading it. It exists and that is all it need do.

    Bill Rechin’s Crock (August 22) is not exactly an anthropomorphized numerals joke. But it’s something in the field of turning numbers into physical objects. It’s cute enough.

    The cannon fires one (ball), then fires two (the numeral). The artillery man says 'I've always wanted to do that.'

    Bill Rechin’s Crock for the 22nd of August, 2015.

    I am curious why the first two panels are duplicates, though. (Look at the hatching on the cannon, or whatever the scribbles are at the bottom of his shirt.) Actually, everything about Crock is a bit mysterious. Cartoonist Bill Rechin died in May of 2011, and his family decided to stop drawing new strips in May of 2012. However, for some reason, reruns were to be distributed for three further years. Still, the strips are dated 2015. I don’t remember seeing them before. Of course, I admit I don’t have many Crock strips committed to memory, but I’d have imagined at least some would have struck me as familiar. In short, there’s a lot I don’t understand about this comic strip.

     
  • Joseph Nebus 3:00 pm on Friday, 21 August, 2015 Permalink | Reply
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    What can you see in the number 585? 


    Joseph Nebus:

    Iva Sallay’s Find The Factors page I’ve mentioned before, since it provides a daily factorization puzzle. That’s a fun recreational mathematics puzzle even if the level 6’s and sometimes level 5’s will sometimes feel impossible. I wanted to point out, though, there’s also talk about the factoring of numbers, and ways to represent that factoring, that’s also interesting and attractive to look at. It can also include neat bits of trivia about numbers and their representation. In this example 585 presents some interesting facets, including several ways that it’s a palindromic number. If you don’t care for, or aren’t interested in, the factoring puzzles you might find it worth visiting for the trivia alone.

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    Originally posted on Find the Factors:

    This week I watched an excellent video titled 5 x 9 is more than 45. Indeed 45 is so much more than simply 5 x 9. Every multiplication fact is much more than that mere fact, but Steve Wyborney used 5 x 9 = 45 in his video… Guess what! 585 is a multiple of 45.

    As I thought about the number 585, I marveled at some of the hidden mysteries this number holds.

    Since 585 is divisible by two different centered square numbers, 5 and 13, I saw that 585 could be represented by this lovely array that has 45 larger squares made up of 13 smaller colorful squares. When you look at the array, do you just see 585 squares or can you see even more multiplication and division facts? If you rotate the array 90 degrees, do the facts change?

    585 Squares-1

    What do you see in this…

    View original 268 more words

     
    • ivasallay 3:10 pm on Friday, 21 August, 2015 Permalink | Reply

      Thanks for the reblog! Also thank you for being one of my most consistent supporters.

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  • Joseph Nebus 3:00 pm on Tuesday, 18 August, 2015 Permalink | Reply
    Tags: detailed balance, , rankings,   

    How Pinball Leagues and Chemistry Work: The Mathematics 


    My love and I play in several pinball leagues. I need to explain something of how they work.

    Most of them organize league nights by making groups of three or four players and having them play five games each on a variety of pinball tables. The groupings are made by order. The 1st through 4th highest-ranked players who’re present are the first group, the 5th through 8th the second group, the 9th through 12th the third group, and so on. For each table the player with the highest score gets some number of league points. The second-highest score earns a lesser number of league points, third-highest gets fewer points yet, and the lowest score earns the player comments about how the table was not being fair. The total number of points goes into the player’s season score, which gives her ranking.

    You might see the bootstrapping problem here. Where do the rankings come from? And what happens if someone joins the league mid-season? What if someone misses a competition day? (Some leagues give a fraction of points based on the player’s season average. Other leagues award no points.) How does a player get correctly ranked?

    (More …)

     
    • sheldonk2014 3:26 pm on Tuesday, 18 August, 2015 Permalink | Reply

      Interesting sounding

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      • Joseph Nebus 9:13 pm on Saturday, 22 August, 2015 Permalink | Reply

        Glad you’re interested. My love and I dropped in the rankings last pinball league meeting, although we were a bit overvalued going into it. So that was just part of the process of getting back to our detailed balance.

        Still, there’s something terrible about putting up my best-ever game on Cirqus Voltaire and coming in third (of four) on that table. On the bright side I put in my best-ever game on Metallica too and came in first. And that was the final game of the night, so I could go out on a high note.

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  • Joseph Nebus 3:23 pm on Sunday, 16 August, 2015 Permalink | Reply
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    Original Problem! Expanding Galaxies and Rates of Change 


    Joseph Nebus:

    Afiq Hatta here presents a nice little problem that mixes geometry and calculus. And it’s inspired by cosmology, to cover an extra subject.

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    Originally posted on The Malaysian Mathematics:

    This post was inspired by an article on cosmology that I read, which looked at the possible models of how the universe expanded. So, I created a problem who’s constraints almost mimic that of three expanding galaxies moving away for each other. The problem is as follows:

    Expanding Galaxies Problem

    Galaxies A and B, B and C, C and A have initial displacements of “a”, “b”, and “c” between them respectively. Given that that the magnitude of the vectors AB, BC, AC are increasing at the same rate, what is the rate of change of the area of the triangle ABC? Express as a function of “phi”, t, and abc.

    Tip: you may find the following formula for the area of a triangle useful!

    CodeCogsEqn (89)

    Where,CodeCogsEqn (90)

    View original

     
    • howardat58 4:08 pm on Sunday, 16 August, 2015 Permalink | Reply

      Hi Joseph
      You need to explain to “The Malaysian Mathematics” that there is a world (universe) of difference between his formulation of the problem, with its constant rates of change for a, b and c, and an expansion, in which the rate of displacement of two points is proportional to the current displacement.
      In the constant case the triangle will eventually become equilateral, in the expansion case the shape of the triangle remains the same and its rate of expansion is proportional to the square of the displacement.
      You can do the math!

      Liked by 1 person

      • afiqhatta 4:48 pm on Sunday, 16 August, 2015 Permalink | Reply

        You are totally correct howardat58 – but it depends on the model of cosmological expansion used! That’s why I said *almost*. I am actually working on a problem which will include a d|AC|/dt = k * AC, and so on. Thank you for your feedback! It is only inspiration, not a following. Sorry once again – I may even edit the point. Of course if you really wanted to go into detail you could include relativity and so forth, depending on what model (Steady state/ etc.) of expansion you wish to cover! Hey if you would like to collaborate….

        Liked by 1 person

        • Joseph Nebus 5:11 am on Tuesday, 18 August, 2015 Permalink | Reply

          I probably ought to have explained more in my introduction that I understood this was a problem inspired by cosmological expansion rather than directly drawn from it. A good problem can come from even an unrealistically loose model.

          Like

    • J. Sánchez 2:52 pm on Wednesday, 19 August, 2015 Permalink | Reply

      Nice post. I like this type of problems because it could make the students more enthusiasts with calculus. I mean, when someone ask to the student: What are you studying in calculus? R/ Something with expanding galaxies.

      Like

      • Joseph Nebus 9:25 pm on Saturday, 22 August, 2015 Permalink | Reply

        I’d like to think so. It’s an imagination-capturing sort of problem, at least, even granting that the relationship to actual galaxy behavior is slight. It’s hard to find problems that are both about stuff people naturally want to know more about and that fit neatly a particular problem to solve.

        Like

  • Joseph Nebus 3:09 pm on Saturday, 15 August, 2015 Permalink | Reply
    Tags: , , rounding, showmanship,   

    Reading the Comics, August 14, 2015: Name-Dropping Edition 


    There have been fewer mathematically-themed comic strips than usual the past week, but they have been coming in yet. This week seems to have included a fair number of name-drops of interesting mathematical concepts.

    David L Hoyt and Jeff Knurek’s Jumble (August 10) name-drops the abacus. It has got me wondering about how abacuses were made in the pre-industrial age. On the one hand they could in principle be made by anybody who has beads and rods. On the other hand, a skillfully made abacus will make the tool so much more effective. Who made and who sold them? I honestly don’t know.

    He needed a partner to build a new abacus business, and his buddy said _____ __ __.

    David L Hoyt and Jeff Knurek’s Jumble for the 10th of August, 2015. The link will likely expire around the 10th of September.

    Mick Mastroianni and Mason Mastroianni’s Dogs of C Kennel (August 11) has Tucker reveal that most of the mathematics he scrawls is just to make his work look harder. I suspect Tucker overdid his performance. My experience is you can get the audience’s eyes to glaze over with much less mathematics on the board.

    Leigh Rubin’s Rubes (August 11) mentions chaos theory. It’s not properly speaking a Chaos Butterfly comic strip. But certainly it’s in the vicinity.

    Zach Weinersmith’s Saturday Morning Breakfast Cereal (August 11) name-drops Banach-Tarski. This is a reference to a famous-in-some-circles theorem, or paradox. The theorem, published in 1924 by Stefan Banach and Alfred Tarski, shows something astounding. It’s possible to take a ball, and disassemble it into a number of pieces. Then, doing nothing more than sliding and rotating the pieces, one can reassemble the pieces to get two balls each with the same volume of the original. If that doesn’t sound ridiculous enough, consider that it’s possible to do this trick by cutting the ball into as few as five pieces. (Four, if you’re willing to exclude the exact center of the original ball.) So you can see why this is called a paradox, and why this joke works for people who know the background.

    Scott Hilburn’s The Argyle Sweater (August 12) illustrates that joke about rounding up the cattle you might have seen going around.

     
    • sarcasticgoat 3:50 pm on Saturday, 15 August, 2015 Permalink | Reply

      Are the scrambled words mathematical? Because I do this kind of thing a lot, and I could only get the first one ‘Query’??

      Like

      • Joseph Nebus 4:47 am on Tuesday, 18 August, 2015 Permalink | Reply

        They’re not mathematical. Well, the third one I could stretch to be mathematical, if I tried, and I might use it if my Reading the Comics post was a little short that week. Only the punch line has a fairly direct mathematical link. The Jumble words don’t tend to be thematically linked. ‘QUERY’ comes up a lot in the puzzles, too.

        Liked by 1 person

  • Joseph Nebus 3:00 pm on Thursday, 13 August, 2015 Permalink | Reply
    Tags: ,   

    At The Home Field 


    There was a neat little fluke in baseball the other day. All fifteen of the Major League Baseball games on Tuesday were won by the home team. This appears to be the first time it’s happened since the league expanded to thirty teams in 1998. As best as the Elias Sports Bureau can work out, the last time every game was won by the home team was on the 23rd of May, 1914, when all four games in each of the National League, American League, and Federal League were home-team wins.

    This produced talk about the home field advantage never having it so good, naturally. Also at least one article claimed the odds of fifteen home-team wins were one in 32,768. I can’t find that article now that I need it; please just trust me that it existed.

    The thing is this claim is correct, if you assume there is no home-field advantage. That is, if you suppose the home team has exactly one chance in two of winning, then the chance of fifteen home teams winning is one-half raised to the fifteenth power. And that is one in 32,768.

    This also assumes the games are independent, that is, that the outcome of one has no effect on the outcome of another. This seems likely, at least as long as we’re far enough away from the end of the season. In a pennant race a team might credibly relax once another game decided whether they had secured a position in the postseason. That might affect whether they win the game under way. Whether results are independent is always important for a probability question.

    But stadium designers and the groundskeeping crew would not be doing their job if the home team had an equal chance of winning as the visiting team does. It’s been understood since the early days of organized professional baseball that the state of the field can offer advantages to the team that plays most of its games there.

    Jack Jones, at Betfirm.com, estimated that for the five seasons from 2010 to 2014, the home team won about 53.7 percent of all games. Suppose we take this as accurate and representative of the home field advantage in general. Then the chance of fifteen home-team wins is 0.537 raised to the fifteenth power. That is approximately one divided by 11,230.

    That’s a good bit more probable than the one in 32,768 you’d expect from the home team having exactly a 50 percent chance of winning. I think that’s a dramatic difference considering the home team wins a bit less than four percent more often than 50-50.

    The follow-up question and one that’s good for a probability homework would be to work out what are the odds that we’d see one day with fifteen home-team wins in the mere eighteen years since it became possible.

     
    • sheldonk2014 6:26 pm on Thursday, 13 August, 2015 Permalink | Reply

      Ok yet another strike for Joseph

      Like

    • ivasallay 7:43 pm on Thursday, 13 August, 2015 Permalink | Reply

      I shared this post on facebook with my son who loves baseball. He then wrote to me, “It’s also assuming not just that there is no homefield advantage, but that the two teams are evenly matched. In 10 of the 15 games, the team with the better overall record won. In only 9 of the games did the winning pitcher have a better ERA than the losing pitcher.”

      Like

      • Joseph Nebus 6:03 am on Saturday, 15 August, 2015 Permalink | Reply

        You’re right, it does assume the teams are equally matched. That’s not a justified assumption except as an admission of ignorance, that we might not know which team is better. (I assume that the full season is enough to indicate the strongest and the weakest teams, although it’s probably not enough to distinguish which is the 15th versus the 16th versus the 17th-strongest teams.)

        Like

    • Angie Mc 2:20 pm on Friday, 14 August, 2015 Permalink | Reply

      Sending this to my college pitcher and will be reading it with my 2 balls players at home today. Very cool, Joseph. Thank you!

      Like

      • Joseph Nebus 6:04 am on Saturday, 15 August, 2015 Permalink | Reply

        I hope he enjoyed, though I’d imagine he had seen some talk about the fluke already.

        Liked by 1 person

        • Angie Mc 5:10 pm on Saturday, 15 August, 2015 Permalink | Reply

          My oldest son is our biggest baseball trivia geek, although we all enjoy cool stuff like this. There’s so much to keep up with so thanks for this one :D

          Like

          • Joseph Nebus 4:59 am on Tuesday, 18 August, 2015 Permalink | Reply

            Oh, good. I’ve had the good fortune the past few years to read up on the history of baseball statistics — it grew up with the organization of baseball — and that’s fascinating stuff. Partly for the history, partly for the mathematics, partly for the sociology of trying to figure out what needs quantifying and how to do that.

            Liked by 1 person

            • Angie Mc 3:26 pm on Tuesday, 18 August, 2015 Permalink | Reply

              Exactly! When my oldest began playing baseball as a kid, I knew nothing about the sport. Nothing. So, I read everything I could get my hands on about it and fell in love, mainly through the history at first. Baseball is so rich!

              Like

              • Joseph Nebus 9:19 pm on Saturday, 22 August, 2015 Permalink | Reply

                I have to admit I’m not much for playing baseball. I would have this problem of not sufficiently holding on to the bat after swinging, although the third baseman was able to jump out of the way of the bat every time. But the lore and the history and the evolution of the game are hard to resist. In short, I’ll buy pretty near anything Peter Morris writes.

                Liked by 1 person

                • Angie Mc 11:56 pm on Saturday, 22 August, 2015 Permalink | Reply

                  LOL! I’m with you about hitting a round ball with a round bat…how can anyone do that?! Morris’s “A Game of Inches” sounds terrific. Have you read it?

                  Like

                  • Joseph Nebus 7:09 pm on Monday, 24 August, 2015 Permalink | Reply

                    It is a terrific book. I’ve read it and keep going back to leaf through it; there’s something fascinating on pretty near every page.

                    For example, you know how it’s legal for the runner to over-run first base? But not second or third? That’s the fossilized result of the decade or so when it was fashionable to play winter baseball on frozen ponds with all the players wearing ice skates.

                    Liked by 1 person

                    • Angie Mc 6:01 am on Tuesday, 25 August, 2015 Permalink | Reply

                      All right, then, it’s official. I need to read this book! Likely after Christmas, before spring training when I need a baseball fix :D

                      Like

                      • Joseph Nebus 11:13 pm on Thursday, 27 August, 2015 Permalink

                        Certainly, yes. It’ll fill that niche very nicely. It’s a thick book but everything in it is half- or one-page chunks, basically. And you can browse and graze rather than reading straight through.

                        Liked by 1 person

                      • Angie Mc 11:19 pm on Thursday, 27 August, 2015 Permalink

                        Perfect! Just added it to my Amazon wish list :D

                        Like

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