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  • Joseph Nebus 6:00 pm on Sunday, 26 March, 2017 Permalink | Reply
    Tags: , , fractions, Moderately Confused, Pardon My Planet, , , , , Todd the Dinosaur   

    Reading the Comics, March 25, 2017: Slow Week Edition 

    Slow week around here for mathematically-themed comic strips. These happen. I suspect Comic Strip Master Command is warning me to stop doing two-a-week essays on reacting to comic strips and get back to more original content. Message received. If I can get ahead of some projects Monday and Tuesday we’ll get more going.

    Patrick Roberts’s Todd the Dinosaur for the 20th is a typical example of mathematics being something one gets in over one’s head about. Of course it’s fractions. Is there anything in elementary school that’s a clearer example of something with strange-looking rules and processes for some purpose students don’t even know what they are? In middle school and high school we get algebra. In high school there’s trigonometry. In high school and college there’s calculus. In grad school there’s grad school. There’s always something.

    Teacher: 'Todd, are you wearing water wings? Why, pray tell?' 'So I can make it to the third grade! We're startin' fractions today and YOU said you had a feeling I was gonna get in over my head.' 'Dang!'

    Patrick Roberts’s Todd the Dinosaur for the 20th of March, 2017. I’ll allow the kids-say-the-darndest-things setup for the strip. I’m stuck on wondering just how much good water wings that size could do. Yes, he’s limited by his anatomy but aren’t we all?

    Jeff Stahler’s Moderately Confused for the 21st is the usual bad-mathematics-of-politicians joke. It may be a little more on point considering the Future Disgraced Former President it names, but the joke is surely as old as politicians and hits all politicians with the same flimsiness.

    John Graziano’s Ripley’s Believe It Or Not for the 22nd names Greek mathematician Pythagoras. That’s close enough to on-point to include here, especially considering what a slow week it’s been. It may not be fair to call Pythagoras a mathematician. My understanding is we don’t know that actually did anything in mathematics, significant or otherwise. His cult attributed any of its individuals’ discoveries to him, and may have busied themselves finding other, unrelated work to credit to their founder. But there’s so much rumor and gossip about Pythagoras that it’s probably not fair to automatically dismiss any claim about him. The beans thing I don’t know about. I would be skeptical of anyone who said they were completely sure.

    Vic Lee’s Pardon My Planet for the 23rd is the usual sort of not-understanding-mathematics joke. In this case it’s about percentages, which are good for baffling people who otherwise have a fair grasp on fractions. I wonder if people would be better at percentages if they learned to say “percent” as “out of a hundred” instead. I’m sure everyone who teaches percentages teaches that meaning, but that doesn’t mean the warning communicates.

    'OK, then let's compromise. I'll be right most of the time - at least 46 percent of the time. And you can be right whenever there is math involved.'

    Vic Lee’s Pardon My Planet for the 23rd of March, 2017. Don’t mind me, I’m busy trying to convince myself the back left leg of that park bench is hidden behind the guy’s leg and not missing altogether and it’s still pretty touch-and-go on that.

    Stephan Pastis’s Pearls Before Swine for the 24th jams a bunch of angle puns into its six panels. I think it gets most of the basic set in there.

    Samson’s Dark Side Of The Horse for the 25th mentions sudokus, and that’s enough for a slow week like this. I thought Horace was reaching for a calculator in the last panel myself, and was going to say that wouldn’t help any. But then I checked the numbers in the boxes and that made it all better.

  • Joseph Nebus 6:00 pm on Tuesday, 25 October, 2016 Permalink | Reply
    Tags: , fractions, , , origami   

    Reading the Comics, October 22, 2016: The Jokes You Can Make About Fractions Edition 

    Last week had a whole bundle and a half of mathematically-themed comics so let me finish off the set. Also let me refresh my appeal for words for my End Of 2016 Mathematics A To Z. There’s all sorts of letters not yet claimed; please think of a mathematical term and request it!

    David L Hoyt and Jeff Knurek’s Jumble for the 19th gives us a chance to do some word puzzle games again. If you like getting the big answer without doing the individual words then pay attention to the blackboard in the comic. Just saying.

    DUEGN O-O-O; NERDT OOO--; NINBUO OO--O-; MUURQO ---OO; The teacher was happy that those who did poorly on the math test were -----------.

    David L Hoyt and Jeff Knurek’s Jumble for the 19th of October, 2016. The link will probably expire in about a month. Have to say, it’s not a big class. I’m not surprised the students are doing well.

    Patrick J Marran’s Francis for the 20th features origami, as well as some of the more famous polyhedrons. The study of what shapes you can make from a flat sheet by origami processes — just folding, no cutting — is a neat one. Apparently origami geometry can be built out of seven axioms. I’m delighted to learn that the axioms were laid out as recently as 1992, with the exception of one that went unnoticed until 2002.

    Gabby describes her shape as an isocahedron, which must be a typo. We all make them. There’s icosahedrons which look like that figure and I’ve certainly slipped consonants around that way.

    I’m surprised and delighted to find there are ways to make an origami icosahedron. Her figure doesn’t look much like the origami icosahedron of those instructions, but there are many icosahedrons. The name just means there are 20 faces to the polyhedron so there’s a lot of room for variants.

    If you were wondering, yes, the Francis of the title is meant to be the Pope. It’s kind of a Pope Francis fan comic. I cannot explain this phenomenon.

    Rick Detorie’s One Big Happy rerun for the 21st retells one of the standard jokes you can always make about fractions. Fortunately it uses that only as part of the setup, which shows off why I’ve long liked Detorie’s work. Good cartoonists — good writers — take a stock joke and add something to make it fit their characters.

    I’ve featured Richard Thompson’s Poor Richard’s Almanac rerun from the 21st before. I’ll surely feature it again. I just like Richard Thompson art like this. This is my dubious inclusion of the essay. In “What’s New At The Zoo” he tosses off a mention of chimpanzees now typing at 120 words per minute. A comic reference to the famous thought experiment of a monkey, or a hundred monkeys, or infinitely many monkeys given typewriters and time to write all the works of literature? Maybe. Or it might just be that it’s a funny idea. It is, of course.

    'Dad, will you check my math homework?' 'Um, it looks like you wrote two different answers to every problem. Shouldn't there be just one?' 'I like to increase my odds.'

    Rick Kirkman and Jerry Scott’s Baby Blues for the 22nd of October, 2016. I’m not quite curious enough to look, but do wonder how far into the comments you have to go before someone slags on the Common Core. But then I would say if Hammy were to write down first an initial-impression guess of about what the answer should be — say, that “37 + 42” should be a number somewhere around 80 — and then an exact answer, then that would be consistent with what I understand Common Core techniques encourage and a pretty solid approach.

    In Rick Kirkman and Jerry Scott’s Baby Blues for the 22nd Hammie offers multiple answers to each mathematics problem. “I like to increase my odds,” he says. For arithmetic problems, that’s not really helping. But it is often useful, especially in modeling complicated systems, to work out multiple answers. If you’re not sure how something should behave, and it’s troublesome to run experiments, then try develop several different models. If the models all describe similar behavior, then, good! It’s reason to believe you’re probably right, or at least close to right. If the models disagree about their conclusions then you need information. You need experimental results. The ways your models disagree can inspire new experiments.

    Mark Leiknes’s Cow and Boy rerun for the 22nd is another with one of the standard jokes you can make about fractions. I suspect I’ve featured this before too, but I quite like Cow and Boy. It’s sad that the strip was cancelled, and couldn’t make a go of it as web comic. I’m not surprised; the strip had so many running jokes it might as well have had a deer and an orca shooting rocket-propelled grenades at new readers. But it’s grand seeing the many, many, many running jokes as they were first established. This is part of the sequence in which Billy, the Boy of the title, discovers there’s another kid named Billy in the class, quickly dubbed Smart Billy for reasons the strip makes clear.

  • Joseph Nebus 6:00 pm on Tuesday, 5 July, 2016 Permalink | Reply
    Tags: , , fractions, , , ,   

    Reading the Comics, July 2, 2016: Ripley’s Edition 

    As I said Sunday, there were more mathematics-mentioning comic strips than I expected last week. So do please read this little one and consider it an extra. The best stuff to talk about is from Ripley’s Believe It Or Not, which may or may not count as a comic strip. Depends how you view these things.

    Randy Glasbergen’s Glasbergen Cartoons for the 29th just uses arithmetic as the sort of problem it’s easiest to hide in bed from. We’ve all been there. And the problem doesn’t really enter into the joke at all. It’s just easy to draw.

    John Graziano’s Ripley’s Believe It Or Not on the 29th shows off a bit of real trivia: that 599 is the smallest number whose digits add up to 23. And yet it doesn’t say what the largest number is. That’s actually fair enough. There isn’t one. If you had a largest number whose digits add up to 23, you could get a bigger one by multiplying it by ten: 5990, for example. Or otherwise add a zero somewhere in the digits: 5099; or 50,909; or 50,909,000. If we ignore zeroes, though, there are finitely many different ways to write a number with digits that add up to 23. This is almost an example of a partition problem. Partitions are about how to break up a set of things into groups of one or more. But in a partition proper we don’t really care about the order: 5-9-9 is as good as 9-9-5. But we can see some minor differences between 599 and 995 as numbers. I imagine there must be a name for the sort of partition problem in which order matters, but I don’t know what it is. I’ll take nominations if someone’s heard of one.

    Graziano’s Ripley’s sneaks back in here the next day, too, with a trivia almost as baffling as the proper credit for the strip. I don’t know what Graziano is getting at with the claim that Ancient Greeks didn’t consider “one” to be a number. None of the commenters have an idea either and my exhaustive minutes of researching haven’t worked it out.

    But I wouldn’t blame the Ancient Greeks for finding something strange about 1. We find something strange about it too. Most notably, of all the counting numbers 1 falls outside the classifications of “prime” and “composite”. It fits into its own special category, “unity”. It divides into every whole number evenly; only it and zero do that, if you don’t consider zero to be a whole number. It’s the multiplicative identity, and it’s the numerator in the set of unit fractions — one-half and one-third and one-tenth and all that — the first fractions that people understand. There’s good reasons to find something exceptional about 1.

    dro-mo for the 30th somehow missed both Pi Day and Tau Day. I imagine it’s a rerun that the artist wasn’t watching too closely.

    Aaron McGruder’s The Boondocks rerun for the 2nd concludes that storyline I mentioned on Sunday about Riley not seeing the point of learning subtraction. It’s always the motivation problem.

  • Joseph Nebus 3:00 pm on Monday, 9 May, 2016 Permalink | Reply
    Tags: , , , fast food, fractions, , ,   

    Reading the Comics, May 6, 2016: Mistakes Edition 

    I knew my readership would drop off after I fell back from daily posting. Apparently it was worse than I imagined and nobody read my little blog here over the weekend. That’s fair enough; I had to tend other things myself. Still, for the purpose of maximizing the number of page views around here, taking two whole days off in a row was a mistake. There’s some more discussed in this Reading The Comics installment.

    Word problems are dull. At least at the primary-school level. There’s all these questions about trains going in different directions or ropes sweeping out areas or water filling troughs. So Aaron McGruder’s Boondocks rerun from the 5th of May (originally run the 22nd of February, 2001) is a cute change. It’s at least the start of a legitimate word problem, based on the ways the recording industry took advantage of artists in the dismal days of fifteen years ago. I’m sure that’s all been fixed by now. Fill in some numbers and the question might interest people.

    Glenn McCoy and Gary McCoy’s The Duplex for the 5th of May is a misunderstanding-fractions joke. I’m amused by the idea of messing up quarter-pound burgers. But it also brings to mind a summer when I worked for the Great Adventure amusement park and got assigned one day as cashier at the Great American Hamburger Stand. Thing is, I didn’t know anything about the stand besides the data point that they probably sold hamburgers. So customers would order stuff I didn’t know, and I couldn’t find how to enter it on the register, and all told it was a horrible mess. If you were stuck in that impossibly slow-moving line, I am sorry, but it was management’s fault; I told them I didn’t know what I was even selling. Also I didn’t know the drink cup sizes so I just charged you for whatever you said and if I gave you the wrong size I hope it was more soda than you needed.

    On a less personal note, I have heard the claim about why one-third-pound burgers failed in United States fast-food places. Several chains tried them out in the past decade and they didn’t last, allegedly because too many customers thought a third of a pound was less than a quarter pound and weren’t going to pay more for less beef. It’s … plausible enough, I suppose, because people have never been good with fractions. But I suspect the problem is more linguistic. A quarter-pounder has a nice rhythm to it. A half-pound burger is a nice strong order to say. A third-pound burger? The words don’t even sound right. You have to say “third-of-a-pound burger” to make it seem like English, and it’s a terribly weak phrase. The fast food places should’ve put their money into naming it something that suggested big-ness but not too-big-to-eat.

    Mark Tatulli’s Heart of the City for the 5th is about Heart’s dread of mathematics. Her expressed fear, that making one little mistake means the entire answer is wrong, is true enough. But how how much is that “enough”? If you add together someting that should be (say) 18, and you make it out to be 20 instead, that is an error. But that’s a different sort of error from adding them together and getting 56 instead.

    And errors propagate. At least they do in real problems, in which you are calculating something because you want to use it for something else. An arithmetic error on one step might grow, possibly quite large, with further steps. That’s trouble. This is known as an “unstable” numerical calculation, in much the way a tin of picric acid dropped from a great height onto a fire is an “unstable” chemical. The error might stay about as large as it started out being, though. And that’s less troublesome. A mistake might stay predictable. The calculation is “stable” In a few blessed cases an error might be minimized by further calculations. You have to arrange the calculations cleverly to make that possible, though. That’s an extremely stable calculation.

    And this is important because we always make errors. At least in any real calculation we do. When we want to turn, say, a formula like πr2 into a number we have to make a mistake. π is not 3.14, nor is it 3.141592, nor is it 3.14159265358979311599796346854418516. Does the error we make by turning π into some numerical approximation matter? It depends what we’re calculating, and how. There’s no escaping error and it might be a comfort to Heart, or any student, to know that much of mathematics is about understanding and managing error.

    The further adventures of Nadine and Nina and Science Friday: 'Does it depress you to know that with the expanding universe and all the countless billions and trillions of other planets, the best-looking men probably aren't even in our galaxy?'

    Joe Martin’s Boffo for the 6th of May, 2016. The link’s already expired, I bet. Yes, the panel did appear on a Sunday.

    Joe Martin’s Boffo for the 6th of May is in its way about the wonder of very large numbers. On some reasonable assumptions — that our experience is typical, that nothing is causing traits to be concentrated one way or another — we can realize that we probably will not see any extreme condition. In this case, it’s about the most handsome men in the universe probably not even being in our galaxy. If the universe is large enough and people common enough in it, that’s probably right. But we likely haven’t got the least handsome either. Lacking reason to suppose otherwise we can guess that we’re in the vast middle.

    David L Hoyt and Jeff Knurek’s Jumble for the 6th of May mentions mathematicians and that’s enough, isn’t it? Without spoiling the puzzle for anyone, I will say that “inocci” certainly ought to be a word meaning something. So get on that, word-makers.

    SMOPT ooo--; ORFPO -o--o; INCOCI o---oo; LAUNAN ooo---. The math teacher was being reprimanded because of his -----------.

    David L Hoyt and Jeff Knurek’s Jumble for the 6th of May, 2016. While ‘ORFPO’ mey not be anything, I believe there should be some company named ‘OrfPro’ that offers some kind of service.

    Dave Blazek’s Loose Parts for the 6th brings some good Venn Diagram humor back to my pages. Good. It’s been too long.

  • Joseph Nebus 3:00 pm on Thursday, 14 April, 2016 Permalink | Reply
    Tags: , flash cards, fractions, , , , , ,   

    Reading the Comics, April 10, 2016: Four-Digit Prime Number Edition 

    In today’s installment of Reading The Comics, mathematics gets name-dropped a bunch in strips that aren’t really about my favorite subject other than my love. Also, I reveal the big lie we’ve been fed about who drew the Henry comic strip attributed to Carl Anderson. Finally, I get a question from Queen Victoria. I feel like this should be the start of a podcast.

    Todd responds to arithmetic flash cards: 'Tater tots! Sloppy Joes! Mac and Cheese!' 'Todd, what are you doing? These are all math!' 'Sorry ... every day at school we have math right before lunch and you told me to say the first thing that pops into my mind!'

    Patrick Roberts’ Todd the Dinosaur for the 6th of April, 2016.

    Patrick Roberts’ Todd the Dinosaur for the 6th of April just name-drops mathematics. The flash cards suggest it. They’re almost iconic for learning arithmetic. I’ve seen flash cards for other subjects. But apart from learning the words of other languages I’ve never been able to make myself believe they’d work. On the other hand, I haven’t used flash cards to learn (or teach) things myself.

    Mom, taking the mathematics book away from Bad Dad: 'I'll take over now ... fractions and long division aren't `scientifically accepted as unknowable`.'

    Joe Martin’s Boffo for the 7th of April, 2016. I bet the link expires in early May.

    Joe Martin’s Boffo for the 7th of April is a solid giggle. (I have a pretty watery giggle myself.) There are unknowable, or at least unprovable, things in mathematics. Any logic system with enough rules to be interesting has ideas which would make sense, and which might be true, but which can’t be proven. Arithmetic is such a system. But just fractions and long division by itself? No, I think we need something more abstract for that.

    Henry is sent to bed. He can't sleep until he reads from his New Math text.

    Carl Anderson’s Henry for the 7th of April, 2016.

    Carl Anderson’s Henry for the 7th of April is, of course, a rerun. It’s also a rerun that gives away that the “Carl Anderson” credit is a lie. Anderson turned over drawing the comic strip in 1942 to John Liney, for weekday strips, and Don Trachte for Sundays. There is no possible way the phrase “New Math” appeared on the cover of a textbook Carl Anderson drew. Liney retired in 1979, and Jack Tippit took over until 1983. Then Dick Hodgins, Jr, drew the strip until 1990. So depending on how quickly word of the New Math penetrated Comic Strip Master Command, this was drawn by either Liney, Tippit, or possibly Hodgins. (Peanuts made New Math jokes in the 60s, but it does seem the older the comic strip the longer it takes to mention new stuff.) I don’t know when these reruns date from. I also don’t know why Comics Kingdom is fibbing about the artist. But then they went and cancelled The Katzenjammer Kids without telling anyone either.

    Eric the Circle for the 8th, this one by “lolz”, declares that Eric doesn’t like being graphed. This is your traditional sort of graph, one in which points with coordinates x and y are on the plot if their values make some equation true. For a circle, that equation’s something like (x – a)2 + (y – b)2 = r2. Here (a, b) are the coordinates for the point that’s the center of the circle, and r is the radius of the circle. This looks a lot like Eric is centered on the origin, the point with coordinates (0, 0). It’s a popular choice. Any center is as good. Another would just have equations that take longer to work with.

    Richard Thompson’s Cul de Sac rerun for the 10th is so much fun to look at that I’m including it even though it just name-drops mathematics. The joke would be the same if it were something besides fractions. Although see Boffo.

    Norm Feuti’s Gil rerun for the 10th takes on mathematics’ favorite group theory application, the Rubik’s Cube. It’s the way I solved them best. This approach falls outside the bounds of normal group theory, though.

    Mac King and Bill King’s Magic in a Minute for the 10th shows off a magic trick. It’s also a non-Rubik’s-cube problem in group theory. One of the groups that a mathematics major learns, after integers-mod-four and the like, is the permutation group. In this, the act of swapping two (or more) things is a thing. This puzzle restricts the allowed permutations down to swapping one item with the thing next to it. And thanks to that, an astounding result emerges. It’s worth figuring out why the trick would work. If you can figure out the reason the first set of switches have to leave a penny on the far right then you’ve got the gimmick solved.

    Pab Sungenis’s New Adventures of Queen Victoria for the 10th made me wonder just how many four-digit prime numbers there are. If I haven’t worked this out wrong, there’s 1,061 of them.

  • Joseph Nebus 3:00 pm on Wednesday, 13 April, 2016 Permalink | Reply
    Tags: , , fractions, , , , ,   

    A Leap Day 2016 Mathematics A To Z: Transcendental Number 

    I’m down to the last seven letters in the Leap Day 2016 A To Z. It’s also the next-to-the-last of Gaurish’s requests. This was a fun one.

    Transcendental Number.

    Take a huge bag and stuff all the real numbers into it. Give the bag a good solid shaking. Stir up all the numbers until they’re thoroughly mixed. Reach in and grab just the one. There you go: you’ve got a transcendental number. Enjoy!

    OK, I detect some grumbling out there. The first is that you tried doing this in your head because you somehow don’t have a bag large enough to hold all the real numbers. And you imagined pulling out some number like “2” or “37” or maybe “one-half”. And you may not be exactly sure what a transcendental number is. But you’re confident the strangest number you extracted, “minus 8”, isn’t it. And you’re right. None of those are transcendental numbers.

    I regret saying this, but that’s your own fault. You’re lousy at picking random numbers from your head. So am I. We all are. Don’t believe me? Think of a positive whole number. I predict you probably picked something between 1 and 10. Almost surely something between 1 and 100. Surely something less than 10,000. You didn’t even consider picking something between 10,012,002,214,473,325,937,775 and 10,012,002,214,473,325,937,785. Challenged to pick a number, people will select nice and familiar ones. The nice familiar numbers happen not to be transcendental.

    I detect some secondary grumbling there. Somebody picked π. And someone else picked e. Very good. Those are transcendental numbers. They’re also nice familiar numbers, at least to people who like mathematics a lot. So they attract attention.

    Still haven’t said what they are. What they are traces back, of course, to polynomials. Take a polynomial that’s got one variable, which we call ‘x’ because we don’t want to be difficult. Suppose that all the coefficients of the polynomial, the constant numbers we presumably know or could find out, are integers. What are the roots of the polynomial? That is, for what values of x is the polynomial a complicated way of writing ‘zero’?

    For example, try the polynomial x2 – 6x + 5. If x = 1, then that polynomial is equal to zero. If x = 5, the polynomial’s equal to zero. Or how about the polynomial x2 + 4x + 4? That’s equal to zero if x is equal to -2. So a polynomial with integer coefficients can certainly have positive and negative integers as roots.

    How about the polynomial 2x – 3? Yes, that is so a polynomial. This is almost easy. That’s equal to zero if x = 3/2. How about the polynomial (2x – 3)(4x + 5)(6x – 7)? It’s my polynomial and I want to write it so it’s easy to find the roots. That polynomial will be zero if x = 3/2, or if x = -5/4, or if x = 7/6. So a polynomial with integer coefficients can have positive and negative rational numbers as roots.

    How about the polynomial x2 – 2? That’s equal to zero if x is the square root of 2, about 1.414. It’s also equal to zero if x is minus the square root of 2, about -1.414. And the square root of 2 is irrational. So we can certainly have irrational numbers as roots.

    So if we can have whole numbers, and rational numbers, and irrational numbers as roots, how can there be anything else? Yes, complex numbers, I see you raising your hand there. We’re not talking about complex numbers just now. Only real numbers.

    It isn’t hard to work out why we can get any whole number, positive or negative, from a polynomial with integer coefficients. Or why we can get any rational number. The irrationals, though … it turns out we can only get some of them this way. We can get square roots and cube roots and fourth roots and all that. We can get combinations of those. But we can’t get everything. There are irrational numbers that are there but that even polynomials can’t reach.

    It’s all right to be surprised. It’s a surprising result. Maybe even unsettling. Transcendental numbers have something peculiar about them. The 19th Century French mathematician Joseph Liouville first proved the things must exist, in 1844. (He used continued fractions to show there must be such things.) It would be seven years later that he gave an example of one in nice, easy-to-understand decimals. This is the number 0.110 001 000 000 000 000 000 001 000 000 (et cetera). This number is zero almost everywhere. But there’s a 1 in the n-th digit past the decimal if n is the factorial of some number. That is, 1! is 1, so the 1st digit past the decimal is a 1. 2! is 2, so the 2nd digit past the decimal is a 1. 3! is 6, so the 6th digit past the decimal is a 1. 4! is 24, so the 24th digit past the decimal is a 1. The next 1 will appear in spot number 5!, which is 120. After that, 6! is 720 so we wait for the 720th digit to be 1 again.

    And what is this Liouville number 0.110 001 000 000 000 000 000 001 000 000 (et cetera) used for, besides showing that a transcendental number exists? Not a thing. It’s of no other interest. And this plagued the transcendental numbers until 1873. The only examples anyone had of transcendental numbers were ones built to show that they existed. In 1873 Charles Hermite showed finally that e, the base of the natural logarithm, was transcendental. e is a much more interesting number; we have reasons to care about it. Every exponential growth or decay or oscillating process has e lurking in it somewhere. In 1882 Ferdinand von Lindemann showed that π was transcendental, and that’s an even more interesting number.

    That bit about π has interesting implications. One goes back to the ancient Greeks. Is it possible, using straightedge and compass, to create a square that’s exactly the same size as a given circle? This is equivalent to saying, if I give you a line segment, can you create another line segment that’s exactly the square root of π times as long? This geometric problem is equivalent to an algebraic one. That problem: can you create a polynomial, with integer coefficients, that has the square root of π as a root? (WARNING: I’m skipping some important points for the sake of clarity. DO NOT attempt to use this to pass your thesis defense without putting those points back in.) We want the square root of π because … well, what’s the area of a square whose sides are the square root of π long? That’s right. So we start with a line segment that’s equal to the radius of the circle and we can do that, surely. Once we have the radius, can’t we make a line that’s the square root of π times the radius, and from that make a square with area exactly π times the radius squared? Since π is transcendental, then, no. We can’t. Sorry. One of the great problems of ancient mathematics, and one that still has the power to attract the casual mathematician, got its final answer in 1882.

    Georg Cantor is a name even non-mathematicians might recognize. He showed there have to be some infinite sets bigger than others, and that there must be more real numbers than there are rational numbers. Four years after showing that, he proved there are as many transcendental numbers as there are real numbers.

    They’re everywhere. They permeate the real numbers so much that we can understand the real numbers as the transcendental numbers plus some dust. They’re almost the dark matter of mathematics. We don’t actually know all that many of them. Wolfram MathWorld has a table listing numbers proven to be transcendental, and the fact we can list that on a single web page is remarkable. Some of them are large sets of numbers, yes, like e^{\pi \sqrt{d}} for every positive whole number d. And we can infer many more from them; if π is transcendental then so is 2π, and so is 5π, and so is -20.38π, and so on. But the table of numbers proven to be irrational is still just 25 rows long.

    There are even mysteries about obvious numbers. π is transcendental. So is e. We know that at least one of π times e and π plus e is transcendental. Perhaps both are. We don’t know which one is, or if both are. We don’t know whether ππ is transcendental. We don’t know whether ee is, either. Don’t even ask if πe is.

    How, by the way, does this fit with my claim that everything in mathematics is polynomials? — Well, we found these numbers in the first place by looking at polynomials. The set is defined, even to this day, by how a particular kind of polynomial can’t reach them. Thinking about a particular kind of polynomial makes visible this interesting set.

    • howardat58 3:26 pm on Wednesday, 13 April, 2016 Permalink | Reply

      I like this stuff. I cut my mathematical teeth on Paul Halmos ” Naive Set Theory”.
      So I thought about “dense set”, “Canto’s middle third”, “countable”, and then realized that U is next. How about “Uncountable” ?


  • Joseph Nebus 3:00 pm on Sunday, 10 April, 2016 Permalink | Reply
    Tags: continued fractions, fractions   

    A Girl’s Thoughts On Continued Fractions 

    I discussed continued fractions recently, and with some controversy. So I imagine people might be interested in another view.

    Friday’s post on the MathsByAGirl blog is on the subject. Continued fractions get more discussion than I offered about how to represent them, and what those representations might tell us.

  • Joseph Nebus 3:00 pm on Friday, 11 March, 2016 Permalink | Reply
    Tags: , fractions, , , ,   

    A Leap Day 2016 Mathematics A To Z: Fractions (Continued) 

    Another request! I was asked to write about continued fractions for the Leap Day 2016 A To Z. The request came from Keilah, of the Knot Theorist blog. But I’d already had a c-word request in (conjecture). So you see my elegant workaround to talk about continued fractions anyway.

    Fractions (continued).

    There are fashions in mathematics. There are fashions in all human endeavors. But mathematics almost begs people to forget that it is a human endeavor. Sometimes a field of mathematics will be popular a while and then fade. Some fade almost to oblivion. Continued fractions are one of them.

    A continued fraction comes from a simple enough starting point. Start with a whole number. Add a fraction to it. 1 + \frac{2}{3}. Everyone knows what that is. But then look at the denominator. In this case, that’s the ‘3’. Why couldn’t that be a sum, instead? No reason. Imagine then the number 1 + \frac{2}{3 + 4}. Is there a reason that we couldn’t, instead of the ‘4’ there, have a fraction instead? No reason beyond our own timidity. Let’s be courageous. Does 1 + \frac{2}{3 + \frac{4}{5}} even mean anything?

    Well, sure. It’s getting a little hard to read, but 3 + \frac{4}{5} is a fine enough number. It’s 3.8. \frac{2}{3.8} is a less friendly number, but it’s a number anyway. It’s a little over 0.526. (It takes a fair number of digits past the decimal before it ends, but trust me, it does.) And we can add 1 to that easily. So 1 + \frac{2}{3 + \frac{4}{5}} means a number a slight bit more than 1.526.

    Dare we replace the “5” in that expression with a sum? Better, with the sum of a whole number and a fraction? If we don’t fear being audacious, yes. Could we replace the denominator of that with another sum? Yes. Can we keep doing this forever, creating this never-ending stack of whole numbers plus fractions? … If we want an irrational number, anyway. If we want a rational number, this stack will eventually end. But suppose we feel like creating an infinitely long stack of continued fractions. Can we do it? Why not? Who dares, wins!

    OK. Wins what, exactly?

    Well … um. Continued fractions certainly had a fashionable time. John Wallis, the 17th century mathematician famous for introducing the ∞ symbol, and for an interminable quarrel with Thomas Hobbes over Hobbes’s attempts to reform mathematics, did much to establish continuous fractions as a field of study. (He’s credited with inventing the field. But all claims to inventing something big are misleading. Real things are complicated and go back farther than people realize, and inventions are more ambiguous than people think.) The astronomer Christiaan Huygens showed how to use continued fractions to design better gear ratios. This may strike you as the dullest application of mathematics ever. Let it. It’s also important stuff. People who need to scale one movement to another need this.

    In the 18th and 19th century continued fractions became interesting for higher mathematics. Continued fractions were the approach Leonhard Euler used to prove that e had to be irrational. That’s one of the superstar numbers of mathematics. Johan Heinrich Lambert used this to show that if θ is a rational number (other than zero) then the tangent of θ must be irrational. This is one path to showing that π must be irrational. Many of the astounding theorems of Srinivasa Ramanujan were about continued fractions, or ideas which built on continued fractions.

    But since the early 20th century the field’s evaporated. I don’t have a good answer why. The best speculation I’ve heard is that the field seems to fit poorly into any particular topic. Continued fractions get interesting when you have an infinitely long stack of nesting denominators. You don’t want to work with infinitely long strings of things before you’ve studied calculus. You have to be comfortable with these things. But that means students don’t encounter it until college, at least. And at that point fractions seem beneath the grade level. There’s a handful of proofs best done by them. But those proofs can be shown as odd, novel approaches to these particular problems. Studying the whole field is hardly needed.

    So, perhaps because it seems like an odd fit, the subject’s dried up and blown away. Even enthusiasts seem to be resigned to its oblivion. Professor Adam Van Tyul, then at Queens University in Kingston, Ontario, composed a nice set of introductory pages about continued fractions. But the page is defunct. Dr Ron Knott has a more thorough page, though, and one with calculators that work well.

    Will continued fractions make a comeback? Maybe. It might take the discovery of some interesting new results, or some better visualization tools, to reignite interest. Chaos theory, the study of deterministic yet unpredictable systems, first grew (we now recognize) in the 1890s. But it fell into obscurity. When we got some new theoretical papers and the ability to do computer simulations, it flowered again. For a time it looked ready to take over all mathematics, although we’ve got things under better control now. Could continued fractions do the same? I’m skeptical, but won’t rule it out.

    Postscript: something you notice quickly with continued fractions is they’re a pain to typeset. We’re all right with 1 + \frac{2}{3 + \frac{4}{5}} . But after that the LaTeX engine that WordPress uses to render mathematical symbols is doomed. A real LaTeX engine gets another couple nested denominators in before the situation is hopeless. If you’re writing this out on paper, the way people did in the 19th century, that’s all right. But there’s no typing it out that way.

    But notation is made for us, not us for notation. If we want to write a continued fraction in which the numerators are all 1, we have a brackets shorthand available. In this we would write 2 + \frac{1}{3 + \frac{1}{4 + \cdots }} as [2; 3, 4, … ]. The numbers are the whole numbers added to the next level of fractions. Another option, and one that lends itself to having numerators which aren’t 1, is to write out a string of fractions. In this we’d write 2 + \frac{1}{3 +} \frac{1}{4 +} \frac{1}{\cdots + }. We have to trust people notice the + sign is in the denominator there. But if people know we’re doing continued fractions then they know to look for the peculiar notation.

    • gaurish 5:09 pm on Friday, 11 March, 2016 Permalink | Reply

      I disagree! Research in field of Continued Fractions never died, so no question of comeback. See following two books:

      (1).Continued Fractions by Aleksandr Yakovlevich Khinchin (1964)
      (2).Neverending Fractions: An Introduction to Continued Fractions by Jonathan Borwein, ‎Alf van der Poorten, ‎Jeffrey Shallit (2014)


      • Joseph Nebus 3:32 am on Monday, 14 March, 2016 Permalink | Reply

        I may be overstating things to say the field’s died. But I don’t remember it ever coming up in my own education, and I can’t — on a quick survey — find evidence of the subject being taught regularly at any of the colleges or universities I’ve had much to do with. It’s mentioned as one of the subjects for a special topics course offered every other year at Michigan State University. But that’s also at the end of the roster, where they usually list the things they’ll get to if there’s time, which there never is.

        And I know these aren’t the only books about continued fractions published recently, but 1964 isn’t all that recent. I am sure good people are finding interesting new results. But the field isn’t thriving the way, say, Monte Carlo methods, or wavelets, or KAM theory are.


        • gaurish 6:12 am on Monday, 14 March, 2016 Permalink | Reply

          Today I just skimmed through a paper on Continued Fractions published in Acta Arithmetica in September 2015. (https://goo.gl/CtXops) It’s recent I guess :-)

          Also, if you haven’t read the 1964 book I suggested in previous comment then you know nothing about continued fractions.

          You probably never dived deep into Number Theory, as I never dived deep into Differential Equations so I don’t know that KAM theory is an active field of research!

          At my Institute (in India), continued fractions are taught in 3rd semester and with decent details. In 2014, a paper on an unsolved problem in continued fractions (Zaremba’s Conjecture) appeared in Annals of Mathematics (http://annals.math.princeton.edu/2014/180-1/p03 )…..

          My whole point was: “If you don’t know something, it doesn’t mean that it doesn’t exist”.


          • Joseph Nebus 7:20 am on Wednesday, 16 March, 2016 Permalink | Reply

            I am happy to take correction. At least, I want to be happy to take correction. You’re right that I don’t know all that’s going on in mathematics — it’s remarkable I know anything that’s going on in mathematics — and I’d be a fool to say courses teaching the subject aren’t there. Thank you for letting me know there’s more in the field than I suspected.


    • KnotTheorist 8:21 pm on Friday, 11 March, 2016 Permalink | Reply

      Thanks for the informative post! I love reading about mathematical history.


  • Joseph Nebus 3:00 pm on Tuesday, 8 March, 2016 Permalink | Reply
    Tags: anime, fractions, , Studio Ghibli   

    Only Fractions 

    My love and I saw Only Yesterday recently. It’s a 1991 Studio Ghibli film, directed by Isao Takahata. It hasn’t had a United States release before, which is a pity; it’s quite good. The movie is about a woman, Taeko, reflecting on her childhood as she considers changing her life. One of the many wonderfully-realized scenes is about ten-year-old Taeko’s struggles with arithmetic. You probably guessed that, as otherwise the movie would seem outside the remit of this blog.

    In the scene Taeko has had a disastrous arithmetic test. Her older sister is trying to coach her through how to divide fractions. It goes lousy. Her older sister insists it’s just a matter of inverting and multiplying. This is a useful tip if you understand how to divide fractions and need to keep straight what you’re doing. If you don’t understand, then it’s whatever the modern equivalent is for instructions on how to set a VCR.

    Taeko tries to understand one problem. \frac{2}{3} \div \frac{4}{1} . She pictures it as an apple and draws a circle, blacking out a third of it. She cuts the rest into four equally-sized pieces and concludes that you could fit six slices into the original apple. Her sister stammers over this and fumes. She declares “that’s multiplication!”. She complains her sister isn’t doing the right thing, she’s not inverting and multiplying. I recognize her sister’s panic. It’s the bluster of someone trying to explain something not actually understood, on watching someone going far off the script.

    The scene’s filled with irony. Taeko has a better understanding of what she’s doing than her sister has, but never knows it. Her sister understands a procedure but not what fractions dividing signifies. She can’t say why one wants to invert anything or multiply something. Taeko knows what the question she’s asked means, but not how to relate that to what she’s asked to do.

    I don’t want to undervalue learning procedures. They’re worth knowing. They are, once you master them, efficient ways to compute. But there are many ways to master a procedure. I can’t believe there is one way to learn anything that works for everyone. One of many challenges teachers face is exploring the different ways their students best learn. Another is getting close enough to how they best learn that most of the students can understand something. It’s a pity when real people akin to Taeko can’t get that little bridge to connect their drawings of an apple to the page of fractions to be worked out.

    • Roger Purves 9:15 pm on Sunday, 27 March, 2016 Permalink | Reply

      to Joseph:

      In the film (*), the problem assigned is to divide 2/3 by 1/4. Takeo is using her drawing of an apple to visualize 2/3 divided by 4. That’s 8/12 divided by 4, or 2/12 = 1/6, Takeo’s answer.

      I rarely “invert and multiply”. Instead, I change to fractions with common denominators; in this case 2/3 to 8/12 an 1/4 to 3/12. How many times does 3/12 go into 8/12? More than twice but less than three times; 2 and 2/3 times.

      I really enjoyed your post, especially the juxtaposition of “invert and multiply” and “whatever the modern equivalent is for instructions on how to set a VCR”. That made me laugh–gallows humor, though.

      Roger Purves

      (*) The site blog.wowzers.com (September 11, 2013) has the relevant clip.


      • Joseph Nebus 5:13 am on Wednesday, 30 March, 2016 Permalink | Reply

        I think moving things to common denominators should help. At least it makes it possible to take the denominators out, and what’s left looks much more like regular old division. But that does also add extra steps to the process, and there is this drive to not do unnecessary steps. But I am biased toward taking extra steps when learning how to do problems.

        Thanks so for the kind words.


  • Joseph Nebus 3:00 pm on Thursday, 25 February, 2016 Permalink | Reply
    Tags: , fractions, , , ,   

    Reading the Comics, February 23, 2016: No Students Resist Word Problems Edition 

    This week Comic Strip Master Command ordered the mention of some of the more familiar bits of mathematical-premise stock that aren’t students resisting word problems. This happens sometimes.

    Rick Stromoski’s Soup to Nutz for the 18th of February finds a fresh joke in the infinite-monkeys problem. Well, it uses a thousand monkeys here, but that hardly matters. If you had one long-enough-lived monkey at the typewriter, in principle, we could expect them to type the works of Shakespeare. It’s how long it takes that changes. In practice, it’s going to be too long to wait for anyway. I wonder if the monkeys will ever get computers to replace their typewriters.

    Carol Lay’s Lay Lines for the 19th finds a fresh joke in Zeno’s Paradoxes. Lay particularly uses the most famous of Zeno’s Paradoxes. That’s the one about not being able to get anywhere because you have to get halfway there first, and halfway to that, in infinite regression. The other of Zeno’s Paradoxes that anyone who hasn’t just read the Wikipedia article on them can remember is Achilles and the Tortoise. It’s the question of how one can catch up to something. By the time you get to where the thing ahead of you is now, it’s gotten farther ahead still. And it does so again, in infinite regression. The third of the Paradoxes is about motion, depicted here as an arrow trying to fly through the air. Allow that speed is the distance travelled versus the time it takes to travel. But suppose time can be divided into infinitesimally tiny units. Then the distance the arrow travels in that time will also be infinitesimally tiny. So how can its speed have any meaningful definition? And the last is a hard-to-follow thing about three rods moving relative one another. I don’t feel confident describing it because I only intermittently feel like I understand what the paradox is getting at. I believe it’s supposed to be a problem with understanding how speeds can add together.

    Anyway, the point of the paradoxes is not something as trite as “silly Ancient Greeks didn’t understand calculus”. They had an awfully good understanding of what makes calculus work. The point is that either space and time are infinitely divisible or else they aren’t. Either possibility has consequences that challenge our intuitions of how space and time should work.

    Dave Blazek’s Loose Parts for the 19th uses scientific notation. It’s a popular way to represent large (and small) numbers. It’s built on the idea that there are two interesting parts to a number: about how big it is, and what its leading values are. We use some base, nearly always 10, raised to a power to represent how big the number is. And we use the rest, a number between 1 and whatever the base is, to represent the leading values. Blazek’s channel 3 x 103 is just channel 3000, though. My satellite TV package has channels numbering from 6 up through 9999, although not all of them. Many are empty. Still, it would be a more excessive number of options if he were on channel 3 x 106, or 3,000,000.

    Russell Myers’s Broom Hilda for the 22nd shows Nerwin trying to learn addition by using a real-world model. I tend to be willing to let people use whatever tool they find works to learn something. But any learning aid has its limits, and trying to get around them can be challenging, or just creepy.

    Dave Whamond’s Reality Check for the 22nd is another version of that rounding-up joke that’s gone around Comic Strip Master Command, and your friends’ Facebook timelines, several times now. Well, I enjoy how suspicious the sheep up front are.

    'Hammie, we do NOT call the police for 'Homework Emergencies'!!' 'But Dad, these fractions are killing me!'

    Rick Kirkman and Jerry Scott’s Baby Blues for the 23rd of February, 2016. You know, that’s an awfully tiny mirror above the keys. There’s no way Wanda and Darryl can even see their whole faces in it.

    Rick Kirkman and Jerry Scott’s Baby Blues for the 23rd I include mostly because I wanted some pictures to include here. But mathematics is always a reliable choice when one needs scary school work to do. And I grant that fraction are particularly unsettling. There is something exotic in being told 1/2 is much bigger than 1/6, when one knows that 2 is so much smaller than 6. And just when one’s gotten comfortable with that, someone has you subtract one fraction from another.

    In the olden days of sailors and shipping, the pay for a ship’s crew would be in shares of the take of the whole venture. The story I have read, but which I am not experienced enough to verify, depends on not understanding fractions. Naive sailors would demand rather than the offered 96th (or whatever) share of the revenues a 100th or 150th or even bigger numbers. Paymasters would pretend to struggle with before assenting to. Perhaps it’s so. Not understanding finance is as old as finance. But it does also feel like a legend designed to answer the question of when will someone need to know mathematics anyway.

    Words: SERDS O O O - -; MENOV O - O - -; GENBIN - O - - O O; and LINKUE - O - O - O. The Professor's explanation of infinity seemed like it was OOOOO-OOOOOO.

    David L Hoyt and Jeff Knurek’s Jumble for the 24th of February, 2016. The link will likely expire in late March. The third scrambled word reveals to me that ‘nebing’ is totally a word that some science fiction project should be able to use.

    David L Hoyt and Jeff Knurek’s Jumble for the 24th is not necessarily a mathematics comic. It could be philosophy or theology or possibly some other fields. Still, I imagine you can have fun working this out even if the final surprise-answer jumped out at me before I looked at the other words.

  • Joseph Nebus 3:00 pm on Tuesday, 15 December, 2015 Permalink | Reply
    Tags: , , , , fractions, , telescopes   

    Reading the Comics, December 13, 2015: More Nearly Like It Edition 

    This has got me closer to the number of comics I like for a Reading the Comics post. There’s two comics already in my file, for the 14th of December, but those can wait until later in the week.

    David L Hoyt and Jeff Knurek’s Jumble for the 11th of December has a mathematics topic. The quotes in the final answer are the hint that it’s a bit of wordplay. The mention of “subtraction” is a hint.

    Words: 'SOLPI', 'NALST', 'BAVEHE', 'CANYLU'. Circled letters, O O - - O, O - - O -, - O - O - O, O - O - - -. The puzzle: To teach subtraction the teacher had a '- - - - - -' - - - -.

    David L Hoyt and Jeff Knurek’s Jumble for the 11th of December, 2015. The link will probably expire in mid-January 2016. Also somehow I’m writing about 2016 being in the imminent future.

    Brian Kliban’s cartoon for the 11th of December (a rerun from who knows when) promises an Illegal Cube Den, and delivers. I’m just delighted by the silliness of it all.

    Greg Evans’s Luann Againn for the 11th of December reprints the 1987 Luann. “Geometric principles of equitorial [sic] astronomical coordinate systems” gets mentioned as a math-or-physics-sounding complicated thing to do. The basic idea is to tell where things are in the sky, as we see them from the surface of the Earth. In an equatorial coordinate system we imagine — we project — where the plane of the equator is, and we can measure things as north or south of that plane. (North is on the same side that the Earth’s north pole is.) That celestial equator is functionally equivalent to longitude, although it’s called declination.

    We also need something functionally equivalent to longitude; that’s called the right ascension. To define that, we need something that works like the prime meridian. Projecting the actual prime meridian out to the stars doesn’t work. The prime meridian is spinning every 24 hours and we can’t publish updated star charts that quickly. What we use as a reference meridian instead is spring. That is, it’s where the path of the sun in the sky crosses the celestial equator in March and the (northern hemisphere) spring.

    There are catches and subtleties, which is why this makes for a good research project. The biggest one is that this crossing point changes over time. This is because the Earth’s orbit around the sun changes. So right ascensions of points change a little every year. So when we give coordinates, we have to say in which system, and which reference year. 2000 is a popular one these days, but its time will pass. 1950 and 1900 were popular in their generations. It’s boring but not hard to convert between these reference dates. And if you need this much precision, it’s not hard to convert between the reference year of 2000 and the present year. I understand many telescopes will do that automatically. I don’t know directly because I have little telescope experience, and I couldn’t even swear I had seen a meteor until 2013. In fairness, I grew up in New Jersey, so with the light pollution I was lucky to see night sky.

    Peter Maresca’s Origins of the Sunday Comics for the 11th of December showcases a strip from 1914. That, Clare Victor Dwiggins’s District School for the 12th of April, 1914, is just a bunch of silly vignettes. It’s worth zooming in to look at. It’s got a student going “figger juggling” and that gives me an excuse to point out the strip to anyone who’ll listen.

    Samson’s Dark Side of the Horse for the 13th of December enters another counting-sheep joke into the ranks. Tying it into angles is cute. It’s tricky to estimate angles by sight. I think people tend to over-estimate how big an angle is when it’s around fifteen or twenty degrees. 45 degrees is easy enough to tell by sight. But for angles smaller than that, I tend to estimate angles by taking the number I think it is and cutting it in half, and I get closer to correct. I’m sure other people use a similar trick.

    Brian Anderson’s Dog Eat Doug for the 13th of December has the dog, Sophie, deploy a lot of fraction talk to confuse a cookie out of Doug. A lot of new fields of mathematics are like that the first time you encounter them. I am curious where Sophie’s reasoning would have led, if not interrupted. How much cookie might she have cadged by the judicious splitting of halves and quarters and, perhaps, eighths and such? I’m not sure where her patter was going.

    Shannon Wheeler’s Too Much Coffee Man for the 13th of December uses the traditional blackboard full of symbols to denote a lot of deeply considered thinking. Did you spot the error?

    • vagabondurges 10:55 pm on Tuesday, 15 December, 2015 Permalink | Reply

      My brain never seemed to jive with the Jumbles, so that may have been the first one I’ve solved. I think I have to stop ignoring them now.


      • Joseph Nebus 4:06 am on Thursday, 17 December, 2015 Permalink | Reply

        Congratulations on solving one. That is how they hook people into solving them compulsively.

        The online version even times you on each word and shows how you compete against the whole online Jumble-solving population.


    • ivasallay 6:12 pm on Wednesday, 16 December, 2015 Permalink | Reply

      I loved the error in the last comic.


    • tziviaeadler 11:22 pm on Wednesday, 16 December, 2015 Permalink | Reply

      LOL, the dark horse ‘angles’ sheep was hilarious


    • sheldonk2014 10:58 am on Sunday, 3 January, 2016 Permalink | Reply

      It’s been happening a lot where I lose someone
      Thanks for taking the time and coming back


  • Joseph Nebus 10:06 pm on Tuesday, 24 February, 2015 Permalink | Reply
    Tags: , Comics Kingdom, employment, fractions, George Stokes, , , , , , Stokes's theorem   

    Reading the Comics, February 24, 2014: Getting Caught Up Edition 

    And now, I think, I’ve got caught up on the mathematics-themed comics that appeared at Comics Kingdom and at Gocomics.com over the past week and a half. I’m sorry to say today’s entries don’t get to be about as rich a set of topics as the previous bunch’s, but on the other hand, there’s a couple Comics Kingdom strips that I feel comfortable using as images, so there’s that. And come to think of it, none of them involve the setup of a teacher asking a student in class a word problem, so that’s different.

    Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. (February 21) tells the old joke about how much of fractions someone understands. To me the canonical version of the joke was a Sydney Harris panel in which one teacher complains that five-thirds of the class doesn’t understand a word she says about fractions, but it’s all the same gag. I’m a touch amused that three and five turn up in this version of the joke too. That probably reflects writing necessity — especially for this B.C. the numbers have to be a pair that obviously doesn’t give you one-half — and that, somehow, odd numbers seem to read as funnier than even ones.

    Bud Fisher’s Mutt and Jeff (February 21) decimates one of the old work-rate problems, this one about how long it takes a group of people to eat a pot roast. It was surely an old joke even when this comic first appeared (and I can’t tell you when it was; Gocomics.com’s reruns have been a mixed bunch of 1940s and 1950s ones, but they don’t say when the original run date was), but the spread across five panels treats the joke well as it’s able to be presented as a fuller stage-ready sketch. Modern comic strips value an efficiently told, minimalist joke, but pacing and minor punch lines (“some men don’t eat as fast as others”) add their charm to a comic.

    (More …)

    • ivasallay 11:22 pm on Wednesday, 25 February, 2015 Permalink | Reply

      My favorites were Mutt and Jeff and Reality Check.
      Has anyone figured out the answer to the Mensa question in Truth Facts?


      • Joseph Nebus 3:23 am on Friday, 27 February, 2015 Permalink | Reply

        I haven’t figured out the Mensa question. I did put together a couple hypotheses but nothing that held up when I looked at it. But I’ve never really gone in for Mensa Test-style questions; they haven’t been fun in ways I quite like.


    • elkement 8:25 pm on Friday, 27 February, 2015 Permalink | Reply

      Really enjoyed Stokes theorem. I wonder how the equivalent explanation for Gauss’ theorem would sound. I can’t find something as graphic as ‘swirly’ for the divergence? ‘Bubbly’ maybe?


      • Joseph Nebus 8:12 pm on Saturday, 28 February, 2015 Permalink | Reply

        You know, that’s a good question, and particularly since it seems like divergence is an easier concept to explain than the swirliness that Stokes’s Theorem requires. But I don’t know a single-word way of describing its source-or-sink nature. Must think about this.

        Liked by 1 person

  • Joseph Nebus 9:08 pm on Monday, 2 February, 2015 Permalink | Reply
    Tags: Alfred S Posamentier, , , , , fractions, Ingmar Lehman,   

    Denominated Mischief 

    I’ve finally got around to reading one of my Christmas presents, Alfred S Posamentier and Ingmar Lehman’s Magnificent Mistakes in Mathematics, which is about ways that mathematical reasoning can be led astray. A lot, at least in the early pages, is about the ways a calculation can be fowled by a bit of carelessness, especially things like dividing by zero, which seems like such an obvious mistake that who could make it once they’ve passed Algebra II?

    They got to a most neat little erroneous calculation, though, and I wanted to share it since the flaw is not immediately obvious although the absurdity of the conclusion drives you to look for it. We begin with a straightforward problem that I think of as Algebra I-grade, though I admit my memories of taking Algebra I are pretty vague these days, so maybe I missed the target grade level by a year or two.

    \frac{3x - 30}{11 - x} = \frac{x + 2}{x - 7} - 4

    Multiply that 4 on the right-hand side by 1 — in this case, by \frac{x - 7}{x - 7} — and combine that into the numerator:

    \frac{3x - 30}{11 - x} = \frac{x + 2 - 4(x - 7)}{x - 7}

    Expand that parentheses and simplify the numerator on the right-hand side:

    \frac{3x - 30}{11 - x} = \frac{3x - 30}{7 - x}

    Since the fractions are equal, and the numerators are equal, therefore their denominators must be equal. Thus, 11 - x = 7 - x and therefore, 11 = 7.

    Did you spot where the card got palmed there?

    • Little Monster Girl 11:13 pm on Monday, 2 February, 2015 Permalink | Reply

      You didn’t do anything to the left side of the equation?


      • Joseph Nebus 10:20 am on Tuesday, 3 February, 2015 Permalink | Reply

        It’s true nothing’s done on the left-hand side, but that isn’t by itself an error. If we start from the assumption that the original equation is true we can manipulate one side, or the other, or both, into a form that’s more convenient without changing whether or not the whole equation is true. The catch is that somewhere in this is a manipulation that doesn’t preserve the truth of the whole thing.

        Liked by 1 person

    • howardat58 11:30 pm on Monday, 2 February, 2015 Permalink | Reply

      Formally, cross multiply is in order. Of course, they don’t call it that these days.


      • Joseph Nebus 10:21 am on Tuesday, 3 February, 2015 Permalink | Reply

        Cross-multiplying ought to give a fair shot at avoiding the error, yeah. But I couldn’t blame someone for seeing an equation of the form a/b = a/d and going right to b = d directly.


    • ivasallay 7:41 am on Tuesday, 3 February, 2015 Permalink | Reply

      Where did (3x – 30)/ (11 – x) = (x + 2)/(x – 7) – 4 come from? It certainly isn’t true for all x.


      • Joseph Nebus 10:23 am on Tuesday, 3 February, 2015 Permalink | Reply

        Well, that’s just a problem to be solved, to find values of x which make it true. It’s just that along the way to finding those x’s, we end up with a conclusion that 11 equals 7.


    • elkement 7:18 pm on Tuesday, 3 February, 2015 Permalink | Reply

      I think the trick is to keep in mind that when the numerator is zero then it does not matter if both denominators are different (as long as they are not equal to zero as well).

      So if x is equal to 10 the equation is true as both sides are equal to zero although the denominators are 1 and -3, respectively.

      The short version is: You must not divide both sides of an equation of zero.


      • Joseph Nebus 11:31 pm on Wednesday, 4 February, 2015 Permalink | Reply

        That’s it exactly, and I’m delighted by the problem since it is one in which the ever-forbidden division by zero is made nicely non-obvious.

        Liked by 1 person

  • Joseph Nebus 8:36 pm on Sunday, 11 January, 2015 Permalink | Reply
    Tags: , , Bon Jovi, , , encryption, fractions, , peace treaties, rot13   

    Reading the Comics, January 11, 2015: Standard Genres And Bloom County Edition 

    I’m still getting back to normal after the Christmas and New Year’s disruption of, well, everything, which is why I’m taking it easy and just doing another comics review. I have to suppose Comic Strip Master Command was also taking it easy over the holidays since most of the subjects are routine genres — word answer problems, mathematics-connected puns, and the like — with the Bloom County reruns the cartoons that give me most to write about. It’s all part of the wondrous cycle of nature; I’m sure there’ll be a really meaty collection of topics along soon.

    Gordon Bess’s Redeye (January 8, originally run August 21, 1968) is an example of the student giving a mischievous answer to a word problem. I feel like I should have a catchy name for this genre, given how much it turns up, but I haven’t got anything good that comes to mind. (I don’t tend to talk about the drawing much in these strips — most of the time it isn’t that important, and comic strips have been growing surprisingly indifferent to drawing — but I did notice while uploading this that Pokey’s stance and expression in the first panel is really quite good. You should be able to open the image in a new tab and see it at its fullest-available 1440-by-431 pixel size and that shows off well the crafting that went into the figure.)

    'If you had six apples and I came along and took five of them, what would that leave you with?' 'A Paleface peace treaty?'

    Gordon Bess’s Redeye for the 8th of January, 2015; originally run the 21st of August, 1968.

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  • Joseph Nebus 9:12 pm on Tuesday, 30 December, 2014 Permalink | Reply
    Tags: , , , expressions, fractions, , , , Rome, state legislatures, Y2K   

    Reading the Comics, December 30, 2014: Surely This Is It For The Year Edition? 

    Well, I thought it’d be unlikely to get too many more mathematics comics before the end of the year, but Comic Strip Master Command apparently sent out orders to clear out the backlog before the new calendar year starts. I think Dark Side of the Horse is my favorite of the strips, blending a good joke with appealing artwork, although The Buckets gives me the most to talk about.

    Greg Cravens’s The Buckets (December 28) is about what might seem only loosely a mathematical topic: that the calendar is really a pretty screwy creation. And it is, as anyone who’s tried to program a computer to show dates has realized. The core problem, I suppose, is that the calendar tries to meet several goals simultaneously: it’s supposed to use our 24-hour days to keep track of the astronomical year, which is an approximation to the cycle of seasons of the year, and there’s not a whole number of days in a year. It’s also supposed to be used to track short-term events (weeks) and medium-term events (months and seasons). The number of days that best approximate the year, 365 and 366, aren’t numbers that lend themselves to many useful arrangements. The months try to divide that 365 or 366 reasonably uniformly, with historial artifacts that can be traced back to the Roman calendar was just an unspeakable mess; and, something rarely appreciated, the calendar also has to make sure that the date of Easter is something reasonable. And, of course, any reforming of the calendar has to be done with the agreement of a wide swath of the world simultaneously. Given all these constraints it’s probably remarkable that it’s only as messed up as it is.

    To the best of my knowledge, January starts the New Year because Tarquin Priscus, King of Rome from 616 – 579 BC, found that convenient after he did some calendar-rejiggering (particularly, swapping the order of February and January), though I don’t know why he thought that particularly convenient. New Years have appeared all over the calendar year, though, with the start of January, the start of September, Christmas Day, and the 25th of March being popular options, and if you think it’s messed up to have a new year start midweek, think about having a new year start in the middle of late March. It all could be worse.

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    • ivasallay 9:01 am on Wednesday, 31 December, 2014 Permalink | Reply

      There are many nights I try to go to sleep thinking about numbers so I definitely related to the Dark Side of the Horse strip.


      • Joseph Nebus 6:12 am on Friday, 2 January, 2015 Permalink | Reply

        I tend to have nights like that more often when I haven’t got any reason to get up early the next day. It’s like my mind is trying to make sure I don’t get a good long sleep any more than absolutely necessary.


    • elkement 4:56 pm on Monday, 5 January, 2015 Permalink | Reply

      I like your comment about making a computer show calendar dates! I still feel that whatever I am working on, it always comes down to fighting with date formats or delimiters or commas, like the German decimal comman versus the English decimal point.


  • Joseph Nebus 11:27 pm on Tuesday, 22 October, 2013 Permalink | Reply
    Tags: fractions, , , , student responses,   

    Split Lines 

    My spouse, the professional philosopher, was sharing some of the engagingly wrong student responses. I hope it hasn’t shocked you to learn your instructors do this, but, if you got something wrong in an amusing way, and it was easy to find someone to commiserate with, yes, they said something.

    The particular point this time was about Plato’s Analogy of the Divided Line, part of a Socratic dialogue that tries to classify the different kinds of knowledge. I’m not informed enough to describe fairly the point Plato was getting at, but the mathematics is plain enough. It starts with a line segment that gets divided into two unequal parts; each of the two parts is then divided into parts of the same proportion. Why this has to be I’m not sure (my understanding is it’s not clear exactly why Plato thought it important they be unequal parts), although it has got the interesting side effect of making exactly two of the four line segments of equal length.

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    • fluffy 7:08 pm on Wednesday, 23 October, 2013 Permalink | Reply

      See also: Douglas Adams’ argument that the universe has zero population.


    • elkement 12:17 pm on Friday, 25 October, 2013 Permalink | Reply

      You have convinced me once more that probability is among the non-intuitive things in mathematics! Nassim Taleb, irreverent trader-turner-philosopher, states that he often found that even professors in statistics make that type of mistake if statistics problems are not presented to them as text book problems, but phrased in natural language in passing. He gave this example: There are two hospitals in a city, a big one and a small one. You know that in one hospitals 60% of all children born in a certain period are boys. Is this rather the big or the small hospital? Taleb said that most ‘experts’ picked the big one though chances for a deviation of the expection value are larger for the small one, thus the smaller sample size.


      • Joseph Nebus 4:47 am on Monday, 28 October, 2013 Permalink | Reply

        Probability is definitely one of the big non-intuitive things (and I’ve been meaning to get a follow-up to this written; it’s been a busy week).

        I’m really startled by the Taleb example, though. I’m curious how the question was presented in the wording since … well, it does feel to me (and to my spouse) that it should be obvious the bigger deviation from the expected average is more likely to happen in the smaller hospital, but perhaps there’s something in the phrasing that throws people. (It’s probably possible to get any response to any probability question by phrasing it right.)


  • Joseph Nebus 5:43 am on Saturday, 18 May, 2013 Permalink | Reply
    Tags: , , , , fractions, ,   

    Reading the Comics, 16 May 2013 

    It’s a good time for another round of comic strip reading, particularly I haven’t had the time to think in detail about all the news in number theory that’s come out this past week, and that I’m not sure whether I should go into explaining arc lengths after I trapped at least one friend into trying to work out the circumference of an ellipse (you can’t do it either, but there are a lot of curves you could). I also notice I’m approaching that precious 10,000th blog hit here, so I can get back to work verifying that law about random data starting with the digit 1.

    Berkeley Breathed’s Bloom County (May 2, rerun) throws up a bunch of mathematical symbols with the intention of producing a baffling result, so that Milo can make a clean getaway from Freida. The splendid thing to me, though, is that Milo’s answer — “log 10 times 10 to the derivative of 10,000” — actually does parse, if you read it a bit charitably. The “log 10” bit we can safely suppose to mean the logarithm base 10, because the strip originally ran in 1981 or so when there was still some use for the common logarithm. These days, we have calculators, and “log” is moving over to be the “natural logarithm”, base e, what was formerly denoted as “ln”.

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    • Solomon Moeuy 4:24 pm on Wednesday, 22 May, 2013 Permalink | Reply

      Thank you so much for the nice little piece dedicated to graphic novels and comic books. I think the thing that I like most relating to graphic novels is their durability. As a professional artist and writer of comics for more than Two decades, I have a tendency to glance at the art initial when searching for a story. Regardless of the topic, a good story is advanced by great graphics that pushes the story forward..


  • Joseph Nebus 8:27 pm on Saturday, 16 March, 2013 Permalink | Reply
    Tags: , fractions, , , ,   

    Reading the Comics, March 12, 2013 

    I’ve got my seven further comic strips with mentions of mathematical topics, so I can preface that a bit with my surprise that at least some of the Gocomics.com comics didn’t bother to mention Pi Day, March 14. It might still be a slightly too much of a This Is Something People Do On The Web observance to be quite sensible for the newspaper comic strips. But there are quite a few strips on Gocomics.com that only appear online, and I thought one of them might.

    (I admit I’m a bit of a Pi Day grouch, on the flimsy grounds that 3/14 is roughly 0.214, which is a rotten approximation to π. But American-style date-writing never gets very good at approximating π. The day-month format used in most of the world offers 22/7 as a less strained Pi Day candidate, except that there’s few schools in session then, wiping out whatever use the day has as a playfully educational event.)

    Gene Weingarten, Dan Weingarten and David Clark’s Barney and Clyde (March 4) introduces a character which I believe is new to the strip, “Norman the math fanatic”. (He hasn’t returned since, as of this writing.) The setup is about the hypothetical and honestly somewhat silly argument about learning math being more important than learning English. I’m not sure I could rate either mathematics or English (or, at least, the understanding of one’s own language) as more important. The panel ends with the traditional scrawl of symbols as shorthand for “this is complicated mathematics stuff”, although it’s not so many symbols and it doesn’t look like much of a problem to me. Perhaps Norman is fanatic about math but doesn’t actually do it very well, which is not something he should be embarrassed about.

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  • Joseph Nebus 11:24 am on Wednesday, 27 February, 2013 Permalink | Reply
    Tags: , fractions, Helsinki,   

    Reading the Comics, February 26, 2013 

    I hit the seven-comics line without quite realizing it, because I’d been dividing my notes between my home computer and one I can access from work. (I haven’t quite taken to writing entries on my iPad, much as I told myself that’d be a great use for it before I bought it, mostly because it’s too annoying to enter all the HTML tags by hand on the iPad keyboard. I’m of the generation that tries to hew its own HTML, even when there’s no benefit to doing that.) This is also skipping a couple strips that just mentioned the kids were in math class because that felt too slight a link to even me.

    Carla Ventresca and Henry Beckett’s On A Claire Day (February 15) discusses the “probability formulas” of a box of chocolates. Distribution functions are just what the name suggests: the set of the possible outcomes of something (like, picking this candy) with the chance of each turning up. It’s useful in simple random-luck problems like gathering candies, but by adding probability distributions to mechanics you create the tool of statistical mechanics, which lets the messy complicated reality of things be treated.

    Pascal Wyse and Joe Berger’s Berger and Wyse (February 18) uses one of the classic motifs of the word problem: fractions as portions of apples, and visualizing fractions by thinking of apple slices. (I tend to eat apples whole, or at least nearly whole, which makes me realize that I probably visualize fractions of apples as a particular instance of fractions rather than as particular versions of apples.)

    Chip Sansom’s The Born Loser (February 21) just shows off Roman numerals and makes fun of the fact they can be misunderstood. But then what can’t?

    Tom Thaves’s Frank and Ernest (February 22) uses the tolerably famous bit of mathematical history about negative numbers being unknown to the ancients and tosses in a joke about the current crisis in the Greek economy so, as ever, don’t read the comments thread.

    William Wilson’s Ordinary Bill (February 22) possibly qualifies for entry into the “silent penultimate panel” family of comic strips (I feel like having significant implied developments in the next-to-the-final panel violates the spirit of the thing but it isn’t my category to define) for a joke about how complicated it is to do one’s taxes. I suspect this is something that’s going to turn up a lot in the coming two months.

    Marc Anderson’s Andertoons (February 24) (I’m wondering whether this or Frank and Ernest gets in here more) pops in with a chalkboard full of math symbols as the way to draw “something incredibly hard to understand”.

    Brian and Ron Boychuk’s The Chuckle Brothers (February 26) has a pie joke that’s so slight I’d almost think they were just angling for the chance for me to notice them. But the name-dropping of the Helsinki Mathematical Institute, and earlier strips with features like references such as to Joseph Henry, make me suspect they’re just enjoying being moderately nerdy. That said, I’m not aware of a specific “Helsinki Mathematical Institute”, although the Rolf Nevanlinna Institute at the University of Helsinki would probably get called something like that. They wouldn’t consider hiring me, anyway.

    • Peter M 9:58 pm on Wednesday, 6 March, 2013 Permalink | Reply

      As a matter of historical interest, when did negative numbers become known, in a separate sense to subtraction of positive numbers?


      • Joseph Nebus 5:00 pm on Friday, 8 March, 2013 Permalink | Reply

        That’s a hard question to answer, not least because it’s really hard to pin down just when anything became known. The earliest generally accepted idea of what negative numbers were seems to be the idea that they’re debts — I suspect this is how people still understand and picture them today, at least to start learning — but I don’t think that is quite a separate sense to subtracting positive integers.

        They went through stages of being slightly more accepted as computational conveniences and useful fictions, with a lot of arguing about the rules by which they would work, and whether such ideas as “a negative times a negative is a positive” even make sense. I think it could be argued it wasn’t until the 19th century and the putting of negative number arithmetic on a pretty sound logical basis that they were quite fully understood as something other than a funny accounting trick.

        This is a neat subject that turns out to be more than I can fully answer right now — I’m away from my primary references — so I should come back to it shortly, when I have the ability.


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