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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Reading The Comics, November 4, 2014: Will Pictures Ever Reappear Edition

I had assumed that at some point the good folks at Comics Kingdom would let any of their cartoonists do a panel that’s got mathematical content relevant enough for me to chat about, but apparently that’s just not happening. So for a third time in a row here’s a set of Gocomics-only comic strips, with reasonably stable links and images I don’t feel the need to include. Enjoy, please.

Fred Wagner’s Animal Crackers (October 26) presents an old joke — counting the number of animals by counting the number of legs and dividing by four — although it’s only silly because it’s hard to imagine a case where it’s easier to count the legs on a bunch of animals than it is to count the animals themselves. But if it’s the case that every animal has exactly four legs, then, there’s what’s called a one-to-one relationship between the set of animals and the set of animal legs: if you have some number of animals you have exactly four times that number of animal legs, and if you have some number of animal legs you have exactly one-fourth that number of animals, and you can count whatever’s the more convenient for you and use that to get what you’re really interested in. Showing such a one-to-one relationship exists between two interesting things can often be a start to doing more interesting problems, especially if you can show that the relationship also preserves some interesting interactions; if you have two ways to work out a problem, you can do the easier one.

Mark Anderson’s Andertoons (October 27) riffs on the place value for numbers written in the familiar Arabic style. As befitting a really great innovation, place value becomes invisible when you’re familiar with it; it takes a little sympathy and imagination to remember the alienness of the idea that a “2” means different things based on how many digits are to the right (or, if it’s a decimal, to the left) of it.

Anthony Blades’s charming Bewley (October 27) has one of the kids insisting that instinct alone is enough to do maths problems. The work comes out disastrously bad, of course, or there’d not be a comic strip. However, my understanding is that people do have some instinctive understanding even of problems that would seem to have little survival application. One test I’ve seen demonstrating this asks people to give, without thinking, their answer to whether a multiplication problem might be right or wrong. It’s pretty quick for most people to say that “7 times 9 equals 12” has to be wrong; to say that “7 times 9 equals 59” is wrong takes longer, and that seems to reflect an idea that 59 is, if not the right answer, at least pretty close to it. There’s an instinctive plausibility at work there and it’s amazing to think people should have that. Zach Weinersmith’s Saturday Morning Breakfast Cereal for October 31 circles around this idea, with one person having little idea what 1,892,491,287 times 7,798,721,415 divided by 82,493,726,631 might be, but being pretty sure that “4” isn’t it.

Saturday Morning Breakfast Cereal (October 30) also contains a mention of “cross products”, which are an interesting thing people learning vectors trip over. A cross product is defined for a pair of three-dimensional vectors, and the interesting thing is it’s a new vector that’s perpendicular to the two vectors multiplied together. The length of the cross product vector depends on the lengths of the two vectors multiplied together and the angle they make; the closer the two vectors multiplied together are, the smaller the cross product is, to the point that the cross product of two parallel vectors has length zero. The closer the two vectors multiplied together are to perpendicular the longer the cross product vector is.

More mysterious: if you swap the first vector and the second vector being cross-multiplied together, you get a cross product that’s the same size but pointing the opposite direction, pointing (say) down instead of up. Cross products have some areas where they’re particularly useful, especially in describing the movement of charged particles in magnetic fields.

(There’s something that looks a lot like the cross product which exists for seven-dimensional vectors, but I’ve never even heard of anyone who had a use for it, so, you don’t need to do anything about it.)

Eric the Circle (November 2), this one by “dDave”, presents the idea that that the points on a line might themselves be miniature Erics the Circle. What a line is made of is again one of those problems that straddles the lines between mathematics and philosophy. It seems to be one of the problems of infinity that Zeno’s Paradoxes outlined so perfectly thousands of years ago. To shorten it to the point it becomes misleading, is a line made up of things that have some width? If they’re infinitesimals, things with no width, then, how can an aggregate of things with no width come to have some width? But if they’re made up of things which have some width, how can there be infinitely many of them fitting into a finite space?

We can form good logical arguments about the convergence of infinite series — lining up, essentially, circles of ever-dwindling but ever-positive sizes so that the pile has a finite length — but that seems to suggest that space has to be made up of intervals of different widths, which seems silly; why couldn’t all the miniature circles be the same? In short, space is either infinitely divisible into identical things, or it is not, and neither one is completely satisfying.

Guy Gilchrist’s Nancy (November 2) uses math homework appearing in the clouds, although that’s surely because it’s easier to draw a division problem than it is to depict an assignment for social studies or English.

Todd Clark’s Lola (November 4) uses an insult-the-minor-characters variant of what seems to be the standard way of explaining fractions to kids, that of dividing a whole thing into smaller pieces and counting the number of smaller pieces. As physical interpretations of mathematical concepts goes I suppose that’s hard to beat.

Reading the Comics, May 13, 2014: Good Class Problems Edition

Someone in Comic Strip Master Command must be readying for the end of term, as there’s been enough comic strips mentioning mathematics themes to justify another of these entries, and that’s before I even start reading Wednesday’s comics. I can’t say that there seem to be any overarching themes in the past week’s grab-bag of strips, but, there are a bunch of pretty good problems that would fit well in a mathematics class here.

Darrin Bell’s Candorville (May 6) comes back around to the default application of probability, questions in coin-flipping. You could build a good swath of a probability course just from the questions the strip implies: how many coins have to come up heads before it becomes reasonable to suspect that something funny is going on? Two is obviously too few; two thousand is likely too many. But improbable things do happen, without it signifying anything. So what’s the risk of supposing something’s up when it isn’t? What’s the risk of dismissing the hints that something is happening?

Mark Anderson’s Andertoons (May 8) is another entry in the wiseacre schoolchild genre (I wonder if I’ve actually been consistent in describing this kind of comic, but, you know what I mean) and suggesting that arithmetic just be done on the computer. I’m sympathetic, however much fun it is doing arithmetic by hand.

Justin Boyd’s Invisible Bread (May 9) is honestly a marginal inclusion here, but it does show a mathematics problem that’s correctly formed and would reasonably be included on a precalculus or calculus class’s worksheets. It is a problem that’s a no-brainer, really, but that fits the comic’s theme of poorly functioning.

Steve Moore’s In The Bleachers (May 12) uses baseball scores and the start of a series. A series, at least once you’re into calculus, is the sum of a sequence of numbers, and if there’s only finitely many of them, here, there’s not much that’s interesting to say. Each sequence of numbers has some sum and that’s it. But if you have an infinite series — well, there, all sorts of amazing things become possible (or at least logically justified), including integral calculus and numerical computing. The series from the panel, if carried out, would come to a pair of infinitely large sums — this is called divergence, and is why your mathematician friends on Facebook or Twitter are passing around that movie poster with a math formula for a divergent series on it — and you can probably get a fair argument going about whether the sum of all the even numbers would be equal to the sum of all the odd numbers. (My advice: if pressed to give an answer, point to the other side of the room, yell, “Look, a big, distracting thing!” and run off.)

Samson’s Dark Side Of The Horse (May 13) is something akin to a pun, playing as it does on the difference between a number and a numeral and shifting between the ways we might talk about “three”. Also, I notice for the first time that apparently the little bird sometimes seen in the comic is named “Sine”, which is probably why it flies in such a wavy pattern. I don’t know how I’d missed that before.

Rick Detorie’s One Big Happy (May 13, rerun) is also a strip that plays on the difference between a number and its representation as a numeral, really. Come to think of it, it’s a bit surprising that in Arabic numerals there aren’t any relationships between the representations for numbers; one could easily imagine a system in which, say, the symbol for “four” were a pair of whatever represents “two”. In A History Of Mathematical Notations Florian Cajori notes that there really isn’t any system behind why a particular numeral has any particular shape, and he takes a section (Section 96 in Book 1) to get engagingly catty about people who do. I’d like to quote it because it’s appealing, in that way:

A problem as fascinating as the puzzle of the origin of language relates to the evolution of the forms of our numerals. Proceeding on the tacit assumption that each of our numerals contains within itself, as a skeleton so to speak, as many dots, strokes, or angles as it represents units, imaginative writers of different countries and ages have advanced hypotheses as to their origin. Nor did these writers feel that they were indulging simply in pleasing pastimes or merely contributing to mathematical recreations. With perhaps only one exception, they were as convinced of the correctness of their explanations as are circle-squarers of the soundness of their quadratures.

Cajori goes on to describe attempts to rationalize the Arabic numerals as “merely … entertaining illustrations of the operation of a pseudo-scientific imagination, uncontrolled by all the known facts”, which gives some idea why Cajori’s engaging reading for seven hundred pages about stuff like where the plus sign comes from.

Reading the Comics, October 25, 2012

As before, this is going to be the comics other than those run through King Features Syndicate, since I haven’t found a solution I like for presenting their mathematics-themed comic strips for discussion. But there haven’t been many this month that I’ve seen either, so I can stick with strips for today at least. (I’m also a little irked that Comics Kingdom’s archives are being shut down — it’s their right, of course, but I don’t like having so many dead links in my old articles.) But on with the strips I have got.

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My Problem With 7

My reposted problem of a couple days ago, about building all the digits of a clock face using exactly three 9’s and simple arithmetic combinations of them, caught in my mind, as these things will sometimes do. The original page missed out on a couple ways of using exactly three 9’s to make a 1, but it’s easy to do. The first thing to wonder about was how big a number could we make using exactly three 9’s? There must be some limit; it’d be absurd to think that we could make absolutely any positive integer with so primitive a tool set — surely 19,686 is out of the realm of attainability — but where is it?

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What Are Numbers Made Of?

To return to my second major theme: my Dearly Beloved told me that I must explain that trick where one adds up the digits of a number and finds out from that whether it’s divisible by 9. I wanted to anyway, but a request like that is irresistible. The answer can be given quickly — and several of my hopefully faithful readers did, in comments, last Friday — but I’d like to take the long way around because I do that and because it lets a lot of other interesting divisibility properties show themselves.

We use ten numerals and the place where we write them to express all the counting numbers out there. We put one of the numerals, such as `2′, in a place which denotes whether we mean to say two tens, or two hundreds, or two millions. That’s a clever tool, and not one inherent to the idea of numbers. We could as easily use different symbols for different magnitudes; the only familiar example of this (in the west) is Roman numerals, where we use I, X, C, and M for increasing powers of ten, and then notice we aren’t really quite sure what to do past M.

The Romans were not very sure either, and individual variations developed when someone found they needed to express an M of M very often. The system has fewer numerals, symbols representing numbers, than ours does, with V and L and D the only additional numerals reasonably common. By the Middle Ages some symbols were improvised to allow for extremely large numbers such as the hundred thousands, and some extra symbols were pulled in for numbers such as 7 or 40, but they have faded to the point of obscurity. This is a numbering system which runs out when the numbers get too large, which seems impossibly limited at first glance. But we haven’t changed much from these times: while we have a numbering system that can, in principle, work with arbitrarily big or tiny numbers, in practice we only use a small range of them. When we turn over arithmetic to computers, in fact, we accept numbering systems which have limits on how big (positive or negative) a number may be, or how close to zero one may work. We accept those limits because of their convenience and are only sometimes annoyed to find, for example, that the spreadsheet trying to calculate a bill has decided we want 0.9999999 of a penny.

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