## Reading the Comics, February 14, 2020: Simple Edition

Greg Evans’s Luann Againn for the 12th features some poor tutoring on Gunther’s part. Usually a person isn’t stuck for what the answer to a problem is; they’re stuck on how to do it correctly. Maybe on how to do it efficiently. But tutoring is itself a skill, and it’s a hard one to learn. We don’t get enough instruction in how to do it.

The problem Luann’s doing is one of simplifying an expression. I remember doing a lot of this, in middle school algebra like that. Simplifying expressions does not change their value; we don’t create new ideas by writing them. So why simplify?

Any grammatically correct expression for a concept may be as good as any other grammatically correct expression. This is as true in writing as it is in mathematics. So what is good writing? There are a thousand right answers. One trait that I think most good writing has is that it makes concepts feel newly accessible. It frames something in a way which makes ideas easier to see. So it is with simplifying algebraic expressions. Finding a version of a formula that makes clearer what you would like to do makes the formula more useful.

Simplifying like this, putting an expression into the fewest number of terms, is common. It typically makes it easier to calculate with a formula. We calculate with formulas all the time. It often makes it easier to compare one formula to another. We compare formulas some of the time. So we practice simplifying like this a lot. Occasionally we’ll have a problem where this simplification is counter-productive and we’d do better to write out something as, to make up an example, $4(x^2 + 2x + 1)^2 + 4(x^2 + 2x + 1) + 1$ instead. Someone who’s gotten good at simplifications, to the point it doesn’t take very much work, is likely to spot cases where one wants to keep part of the expression un-simplified.

Chen Weng’s Messycow Comics for the 13th starts off with some tut-tutting of lottery players. Objectively, yes, money put on a lottery ticket is wasted; even, for example, pick-three or pick-four daily games are so unlikely to pay any award as to be worth it. But I cannot make myself believe that this is necessarily a more foolish thing to do with a couple dollars than, say, buying a candy bar or downloading a song you won’t put on any playlists.

And as the Cow points out, the chance of financial success in art — in any creative field — is similarly ridiculously slight. Even skilled people need a stroke of luck to make it, and even then, making it is a marginal matter. (There is a reason I haven’t quit my job to support myself by blog-writing.) People are terrible at estimating probabilities, especially in situations that are even slightly complicated.

Hector D. Cantü and Carlos Castellanos’s Baldo for the 14th just has Gracie very enthusiastic for arithmetic class. It’s a cute bit.

And now I’m all caught up. Please check in this link next week as I read the comics for their mathematics content some more.

## Reading the Comics, November 7, 2019: The Casual Mentions Edition

As will sometimes happen it’s inconvenient for met to write up a paragraph or two on the particularly mathematically significant comic strips of the past week. Let me here share the comics that just mentioned mathematics, then, and save the heavy stuff for a bit later on.

Bill Watterson’s Calvin and Hobbes repeat for the 5th has Calvin seeking help with an arithmetic problem.

Greg Evans’s Luann Againn for the 6th — a repeat of Luann from the 6th of November, 1991 — sees Gunther thinking out walking to Luann’s house in the language of a word problem. It doesn’t help clear his thoughts.

Jason Chatfield’s Ginger Meggs for the 7th is the sudoku joke for the week.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 7th uses a knowledge of mathematics as shorthand for general knowledge. The strip does misspell “Pythagorean”. This could be a slip on the cartoonists’ part that got past their editors too. Or it could be an extra joke on how often the know-it-all, really, does not. He’ll just talk a confident game long after everyone else has stopped really listening. (I’m a recovering know-it-all myself. I know how our kind thinks.) Or it might be trolling know-it-alls into correcting them.

And this covers things through to Friday’s comics. I write this not having had the chance to read Saturday’s yet. When I do, and when I have the whole week’s strips to discuss, I’ll have it at this link. Furthermore, this week sees the last quarter of the Fall 2019 A to Z under way. I’m excited to learn what I’m doing for the letter ‘U’ also.

## Reading the Comics, September 20, 2019: Quarters and Bunnies Edition

Norm Feuti’s Gil did not last long enough in syndication. This is a shame. The characters were great, the humor in a mode I like, and young Gil’s fascination with shows about the paranormal was eerily close to my own young self. But it didn’t last; my understanding is newspapers were reluctant to bring in a comic strip starring an impoverished family. This is a many-faceted shame, not least because the eternal tension between Gil’s fantasy life and his reality made it one of the few strips to reproduce the most vital element of Calvin and Hobbes. But Feuti decided to resume drawing Sunday strips, and I choose to include that in my Reading the Comics reading, because this is my blog and I can make the rules here, at least.

So here’s Norm Feuti’s Gil for the 15th. A couple days ago I saw someone amazed at finally learning where sunflower seeds come from. They’re the black part in the center of a sunflower, the part that makes the big yellow flower stand out in such contrast. People were giving the poster a hard time, asking, where did he think they came from? And the answer is just, he hadn’t thought about it. Why would he? It’s quite reasonable to go through life never encountering a sunflower seed except as a snack or as part of bird or squirrel food. Where on the sunflower plant it’d even be just doesn’t come up. If you want to make this a dire commentary on society losing its sense of where things come from, all right, I won’t stop you. But I think it’s more that there are a billion things to notice in the world, and so many things have names that are fanciful or allusive or ironic, that it’s normal not to realize that a phrase might literally represent its content.

So Gil having so associated a quarter with 25 cents, rather than one-fourth of a something, makes sense to me. (Especially given, as noted, that he and his mother are poor, and so he grows up attentive to cash.)

Isaac Asimov, prolific writer of cozy mysteries, had one short story built on the idea that a person might misremember 5:50, seen on a digital clock, as half-past five. I mention this to show how the difference between a quarter of a hundred of things, and the quarter of sixty things, will get mixed together.

Greg Evans’s Luann Againn for the 15th sees Luann struggling with algebra. And thinking of ways to at least get the answers. One advantage mathematics instructors have which many other subjects don’t is that you can create more problems easily. If for some reason $\frac{7x + 3}{x - 3}$ isn’t usable anymore, you can make it $\frac{7x + 5}{x - 5}$ and still be testing the same skills. But if you want to (as is reasonable) stick to what’s in a published text, yeah, you’re vulnerable to this.

And you can’t always just change a problem arbitrarily. For example, the expression in the second panel of the top row — $\frac{x^2 - 5x + 6}{x^2 + 5x + 4}$ — I notice factors into $\frac{(x - 3)(x - 2)}{(x + 4)(x + 1)}$. I don’t know the objective of Luann’s homework, but it would probably be messed up if the problem were just changed to $\frac{x^2 - 5x + 8}{x^2 + 5x + 3}$. Not that this couldn’t be worked, but that the work would involve annoying and complicated expressions instead of nice whole numbers or reasonable fractions.

Paul Trap’s Thatababy for the 15th presents Thatabay’s first counting-exponentially book, with the number of rabbits doubling every time. I admire the work Trap put in to drawing — in what we see here — 255 bunnies. I’m trusting there’s 128 in the last bunny panel; I’m not counting. At any rate he drew enough bunnies to not make it obvious to me where he repeats figures.

The traditional ever-increasing bunny spiral is the Fibonacci series. But in that, each panel would on average have only about three-fifths more bunnies than the one before it. That’s good, but it isn’t going to overwhelm as fast as the promise of 256 bunnies on the next page will.

Eric the Circle for the 17th, this by Griffenetsabine, has come up here before. That was back in October of 2013, though, so I don’t blame you for forgetting.

The “dual” here is a mathematical term. Many mathematical things have duals. Polyhedrons have a commonly defined dual shape, though. Start with a polyhedron like, oh, the cube. The dual is a new polyhedron. The vertices of the dual are at the centers of the faces of the original polyhedron. And if two faces of the original polyhedron meet at an edge, then there’s an edge connecting the vertices at the centers of those faces. If several faces meet at a vertex in the original polyhedron, then in the dual there’s a face connecting the vertices dual to the original faces. Work all this out and you get, as you might expect, that the shape that’s dual to a cube is the octahedron we’re told just walked into the bar. The dual to the octahedron, meanwhile … well, that is a cube, which is nice and orderly. You might get a bit of a smile working out what the dual to a tetrahedron is.

Duals are useful, generically, because usually if you can prove something about a dual then you can prove it about the original thing. And we may find that something is easier to prove for the dual than for the original. This isn’t guaranteed, especially for geometric shapes like this, where it’s hard to say that either shape is harder to work with than the other. But it’s one of the tools we have to try sliding between the problem we need to do and the problem we can do.

Olivia Jaimes’s Nancy for the 17th has claims about the usefulness of arithmetic. And Nancy skeptical of them, as you expect for a kid facing mathematics in a comic strip. I admit I’ve never needed to do much arithmetic when I cooked. The most would be figuring out how to adjust the cooking time when two things need very different temperatures. But I always do that by winging it. Now I’m curious whether there are good references for suggested alternate times.

I expect to have another Reading the Comics post here, on Monday. The A to Z series should pick up on Tuesday And I’m still glad for suggestions for the letters I through N. Thank you for reading.

## Reading the Comics, April 10, 2019: Grand Avenue and Luann Want My Attention Edition

So this past week has been a curious blend for the mathematically-themed comics. There were many comics mentioning some mathematical topic. But that’s because Grand Advenue and Luann Againn — reprints of early 90s Luann comics — have been doing a lot of schoolwork. There’s a certain repetitiveness to saying, “and here we get a silly answer to a story problem” four times over. But we’ll see what I do with the work.

Mark Anderson’s Andertoons for the 7th is Mark Anderson’s Andertoons for the week. Very comforting to see. It’s a geometry-vocabulary joke, with Wavehead noticing the similar ends of some terms. I’m disappointed that I can’t offer much etymological insight. “Vertex”, for example, derives from the Latin for “highest point”, and traces back to the Proto-Indo-European root “wer-”, meaning “to turn, to bend”. “Apex” derives from the Latin for “summit” or “extreme”. And that traces back to the Proto-Indo-European “ap”, meaning “to take, to reach”. Which is all fine, but doesn’t offer much about how both words ended up ending in “ex”. This is where my failure to master Latin by reading a teach-yourself book on the bus during my morning commute for three months back in 2002 comes back to haunt me. There’s probably something that might have helped me in there.

Mac King and Bill King’s Magic in a Minute for the 7th is an activity puzzle this time. It’s also a legitimate problem of graph theory. Not a complicated one, but still, one. Graph theory is about sets of points, called vertices, and connections between points, called edges. It gives interesting results for anything that’s networked. That shows up in computers, in roadways, in blood vessels, in the spreads of disease, in maps, in shapes.

One common problem, found early in studying graph theory, is about whether a graph is planar. That is, can you draw the whole graph, all its vertices and edges, without any lines cross each other? This graph, with six vertices and three edges, is planar. There are graphs that are not. If the challenge were to connect each number to a 1, a 2, and a 3, then it would be nonplanar. That’s a famous non-planar graph, given the obvious name K3, 3. A fun part of learning graph theory — at least fun for me — is looking through pictures of graphs. The goal is finding K3, 3 or another one called K5, inside a big messy graph.

Mike Thompson’s Grand Avenue for the 8th has had a week of story problems featuring both of the kid characters. Here’s the start of them. Making an addition or subtraction problem about counting things is probably a good way of making the problem less abstract. I don’t have children, so I don’t know whether they play marbles or care about them. The most recent time I saw any of my niblings I told them about the subtleties of industrial design in the old-fashioned Western Electric Model 2500 touch-tone telephone. They love me. Also I’m not sure that this question actually tests subtraction more than it tests reading comprehension. But there are teachers who like to throw in the occasional surprisingly easy one. Keeps students on their toes.

Greg Evans’s Luann Againn for the 10th is part of a sequence showing Gunther helping Luann with her mathematics homework. The story started the day before, but this was the first time a specific mathematical topic was named. The point-slope form is a conventional way of writing an equation which corresponds to a particular line. There are many ways to write equations for lines. This is one that’s convenient to use if you know coordinates for one point on the line and the slope of the line. Any coordinates which make the equation true are then the coordinates for some point on the line.

Doug Savage’s Savage Chickens for the 10th tosses in a line about logical paradoxes. In this case, using a classic problem, the self-referential statement. Working out whether a statement is true or false — its “truth value” — is one of those things we expect logic to be able to do. Some self-referential statements, logical claims about themselves, are troublesome. “This statement is false” was a good one for baffling kids and would-be world-dominating computers in science fiction television up to about 1978. Some self-referential statements seem harmless, though. Nobody expects even the most timid world-dominating computer to be bothered by “this statement is true”. It takes more than just a statement being about itself to create a paradox.

And a last note. The blog hardly needs my push to help it out, but, sometimes people will miss a good thing. Ben Orlin’s Math With Bad Drawings just ran an essay about some of the many mathematics-themed comics that Hilary Price and Rina Piccolo’s Rhymes With Orange has run. The comic is one of my favorites too. Orlin looks through some of the comic’s twenty-plus year history and discusses the different types of mathematical jokes Price (with, in recent years, Piccolo) makes.

Myself, I keep all my Reading the Comics essays at this link, and those mentioning some aspect of Rhymes With Orange at this link.

## Reading the Comics, January 13, 2019: January 13, 2019 Edition

I admit I’m including a fairly marginal strip in this, just so I can have the fun of another single-day edition. What can I say? I can be easily swayed by silly things. Also, somehow, all four strips today have circumstances where one might mistake them for reruns. Let’s watch.

Bill Amend’s FoxTrot for the 13th is wordplay, mashing up ‘cell division’ with ‘long division’. As you might expect from Bill Amend — who loves sneaking legitimate mathematics and physics in where it’s not needed — Paige’s long cell division is a legitimate one. If you’d like a bit of recreational mathematics fun, you can figure out which microscopic organisms correspond to which numerals. The answer is also the Featured Comment on the page, at least as I write this. So if you need an answer, or you want to avoid having the answer spoiled, know what’s there.

Greg Evans’s Luann Againn for the 13th is the strip of most marginal relevance here. Part of Luann’s awful ay is a mathematics test. The given problems are nothing particularly meaningful. There is the sequence ‘mc2’ in the problem, although written as $m^c 2$. There’s also a mention of ‘googleplex’, which when the strip was first published in 1991 was nothing more than a misspelling of the quite large number. (‘Googol’ is the number; ‘Google’ a curious misspelling. Or perhaps a reversion. The name was coined in 1938 by Milton Sirotta. Sirotta was seven years old at the time. I accept that it is at least possible Sirotta was thinking of the then-very-popular serial-comic strip Barney Google, and that his uncle Edward Kasner, who brought the name to mathematics, wrote it down wrong.) And that carries with it the connotation that big numbers are harder than small numbers. This is … kind of true. At least, long numbers are more tedious than short numbers. But you don’t really do different work, dividing 1428 by 7, than you do dividing 147 by 7. It’s just longer. “Hard” is a flexible idea.

Mac King and Bill King’s Magic in a Minute for the 13th felt like a rerun to me. It took a bit of work to find, but yeah, it was. The strip itself, as presented, is new. But the same neat little modular-arithmetic coincidence was used the 31st of July, 2016.

Mathematics on clock faces is often used as a way to introduce modular arithmetic, a variation on arithmetic with only finitely many integers. This can help, if you’re familiar with clock faces. Like regular arithmetic, modular arithmetic can form a group and a ring. Clock faces won’t give you a group or ring, not unless you replace the number before ‘1’ with a ‘0’. To be a group, you need a collection of items, and a binary operation on the items. This operation we often think of as either addition or multiplication, depending on what makes sense for the problem. To be a ring, you need two binary operations, which interact by a distributive law. So the operations are often matched to addition and multiplication. Modular arithmetic is fun, yes. It’s also useful, not just as a way to do something like arithmetic that’s different. Many schemes for setting up checksums, quick and easy tests against data entry errors, rely on modular arithmetic on the data. And many schemes for generating ‘random’ numbers are built on finding multiplicative inverses in modular arithmetic. This isn’t truly random, of course. But you can look at a string of digits and not see any clear patterns. This is often as close to random as you need.

Rick DeTorie’s One Big Happy for the 13th is mostly a bunch of complaints the old always have against the young. Well, the complaint about parallel parking I haven’t seen before. But the rest are common enough. Featured in it is a complaint that the young can’t do arithmetic. I’m not sure there was ever a time that the older generation thought the young were well-trained in arithmetic. Nor that there was ever a time that the current educational vogue wasn’t blamed for destroying a generation’s ability to calculate. I’m sure there are better and worse ways to teach calculation. But I suspect any teaching method will fall short of addressing a couple issues. One is that people over-rate their own competence and under-rate other’s competence. So the older generation will see itself as having got the best possible arithmetic education and anything that’s different is a falling away. And another is that people get worse at stuff they don’t think is enjoyable or don’t have to do a lot. If you haven’t got a use for the fact, or an appreciation for the beauty in it, three times six is a bit of trivia, and not one that inspires much conversation when shared.

There’s more comics with something of a mathematical theme that got published last week. When I get to them the essays should be at this link.

## Reading the Comics, June 1, 2018: His First Name Is Tom For What That’s Worth Edition

And now I’ve got caught up with last week’s comics. I can get to readying for this coming Sunday looking at … so far … nine comic strips that made the preliminary cut. Whimper.

This time the name does mean something.

Thaves’s Frank and Ernest for the 31st complains about not being treated as a “prime number”. There’s a lot of linguistic connotation gone into this strip. The first is the sense that to be a number is to be stripped of one’s humanity, to become one of a featureless horde. Each number is unique, of course; Iva Sallay’s Find the Factors page each day starts with some of the features of each whole number in turn. But one might look at, oh, 84,644 and not something very different from 84,464.

And yet there’s the idea that there are prime numbers, celebrities within the anonymous counting numbers. The name even says it; a prime something is especially choice. And we speak of prime numbers as somehow being the backbone of numbers. This reflects that we find unique factorizations to be a useful thing to do. But being a prime number doesn’t make a number necessarily better. There are reasons most (European) currencies, before decimalization, divided their currency unit into 20 parts of 12 parts each. And nobody divided them into 19 parts of 13 parts each. As often happens, whether something is good depends on what you’re hoping it’s good for.

Nate Fakes’s Break of Day for the 1st of June is more or less the anthropomorphized numerals installment for the week. It’s also a bit of wordplay, so, good on them. There’s not so many movies about mathematics. Darren Aronofsky’s Pi, Ron Howard’s A Beautiful Mind, and Theodore Melfi’s Hidden Figures are the ones that come to mind, at least in American cinema. And there was the TV detective series Numbers. It seems odd that there wasn’t, like, some little studio prestige thing where Paul Muni played Évariste Galois back in the day. But a lot of the mathematical process isn’t cinematic. People scribbling notes, typing on a computer, or arguing about something you don’t understand are all hard to make worth watching. And the parts that anyone could understand — obsession, self-doubt, arguments over priority, debates about implications — are universal to any discovery or invention. Note that the movies listed are mostly about people who happen to be doing mathematics. You could change the specialties to, say, chemical engineering without altering the major plot beats. Well, Pi would need more alteration. But you could make it about any process that seems to offer reliable forecasting in a new field.

Greg Evans’s Luann Againn for the 1st takes place in mathematics class. The subject doesn’t matter for the joke. It could be anything that doesn’t take much word-balloon space but that someone couldn’t bluff their way through.

Ted Shearer’s Quincy for the 7th of April, 1979 has Quincy thinking what he’ll do with his head for figures. He sees accounting as plausible. Good for him. Society always needs accountants. And they probably do more of society’s mathematics than the mathematicians do.

Bill Abbott’s Spectickles for the 1st features the blackboard-full-of-mathematics to represent the complicated. It shows off the motif that an advanced mathematical formula will be a long and complicated one. This has good grounds behind it. If you want to model something interesting that hasn’t been done before, chances are it’s because you need to consider many factors. And trying to represent them will be clumsily done. It takes reflection and consideration and, often, new mathematical tools to make a formula pithy. Famously, James Clerk Maxwell introduced his equations about electricity and magnetism as a set of twenty equations. By 1873 Maxwell, making some use of quaternions, was able to reduce this to eight equations. Oliver Heaviside, in the late 19th century, used the still-new symbols of vector mechanics. This let him make an attractive quartet. We still see that as the best way to describe electromagnetic fields. As with writing, much of mathematics is rewriting.

## Reading the Comics, February 11, 2018: February 11, 2018 Edition

And it’s not always fair to say that the gods mock any plans made by humans, but Comic Strip Master Command has been doing its best to break me of reading and commenting on any comic strip with a mathematical theme. I grant that I could make things a little easier if I demanded more from a comic strip before including it here. But even if I think a theme is slight that doesn’t mean the reader does, and it’s easy to let the eye drop to the next paragraph if the reader does think it’s too slight. The anthology nature of these posts is part of what works for them. And then sometimes Comic Strip Master Command sends me a day like last Sunday when everybody was putting in some bit of mathematics. There’ll be another essay on the past week’s strips, never fear. But today’s is just for the single day.

Susan Camilleri Konar’s Six Chix for the 11th illustrates the Lemniscate Family. The lemniscate is a shape well known as the curve made by a bit of water inside a narrow tube by people who’ve confused it with a meniscus. An actual lemniscate is, as the chain of pointing fingers suggests, a figure-eight shape. You get — well, I got — introduced to them in prealgebra. They’re shapes really easy to describe in polar coordinates but a pain to describe in Cartesian coordinates. There are several different kinds of lemniscates, each satisfying slightly different conditions while looking roughly like a figure eight. If you’re open to the two lobes of the shape not being the same size there’s even a kind of famous-ish lemniscate called the analemma. This is the figure traced out by the sun if you look at its position from a set point on the surface of the Earth at the same clock time each day over the course of the year. That the sun moves north and south from the horizon is easy to spot. That it is sometimes east or west of some reference spot is a surprise. It shows the difference between the movement of the mean sun, the sun as we’d see it if the Earth had a perfectly circular orbit, and the messy actual thing. Dr Helmer Aslasken has a fine piece about this, and how it affects when the sun rises earliest and latest in the year.

There’s also a thing called the “polynomial lemniscate”. This is a level curve of a polynomial. That is, what are all the possible values of the independent variable which cause the polynomial to evaluate to some particular number? This is going to be a polynomial in a complex-valued variable, in order to get one or more closed and (often) wriggly loops. A polynomial of a real-valued variable would typically give you a boring shape. There’s a bunch of these polynomial lemniscates that approximate the boundary of the Mandelbrot Set, that fractal that you know from your mathematics friend’s wall in 1992.

Mark Anderson’s Andertoons took care of being Mark Anderson’s Andertoons early in the week. It’s a bit of optimistic blackboard work.

Lincoln Pierce’s Big Nate features the formula for calculating the wind chill factor. Francis reads out what is legitimately the formula for estimating the wind chill temperature. I’m not going to get into whether the wind chill formula makes sense as a concept because I’m not crazy. The thinking behind it is that a windless temperature feels about the same as a different temperature with a particular wind. How one evaluates those equivalences offers a lot of room for debate. The formula as the National Weather Service, and Francis, offer looks frightening, but isn’t really hard. It’s not a polynomial, in terms of temperature and wind speed, but it’s close to that in form. The strip is rerun from the 15th of February, 2009, as Lincoln Pierce has had some not-publicly-revealed problem taking him away from the comic for about a month and a half now.

Jim Scancarelli’s Gasoline Alley included a couple of mathematics formulas, including the famous E = mc2 and the slightly less famous πr2, as part of Walt Wallet’s fantasy of advising scientists and inventors. (Scientists have already heard both.) There’s a curious stray bit in the corner, writing out 6.626 x 102 x 3 that I wonder about. 6.626 is the first couple digits of Planck’s Constant, as measured in Joule-seconds. (This is h, not h-bar, I say for the person about to complain.) It’d be reasonable for Scancarelli to have drawn that out of a physics book or reference page. But the exponent is all wrong, even if you suppose he mis-wrote 1023. It should be 6.626 x 10-34. So I don’t know whether Scancarelli got things very garbled, or if he just picked a nice sciencey-looking number and happened to hit on a significant one. (There’s enough significant science numbers that he’d have a fair chance of finding something.) The strip is a reprint from the 4th of February, 2007, as Jim Scancarelli has been absent for no publicly announced reason for four months now.

Greg Evans and Karen Evans’s Luann is not perfectly clear. But I think it’s presenting Gunther doing mathematics work to support his mother’s contention that he’s smart. There’s no working out what work he’s doing. But then we might ask how smart his mother is to have made that much food for just the two of them. Also that I think he’s eating a potato by hand? … Well, there are a lot of kinds of food that are hard to draw.

Greg Evans’s Luann Againn reprints the strip from the 11th of February (again), 1990. It mentions as one of those fascinating things of arithmetic an easy test to see if a number’s a multiple of nine. There are several tricks like this, although the only ones anybody can remember are finding multiples of 3 and finding multiples of 9. Well, they know the rules for something being a multiple of 2, 5, or 10, but those hardly look like rules, and there’s no addition needed. Similarly with multiples of 4.

Modular arithmetic underlies all these rules. Once you know the trick you can use it to work out your own add-up-the-numbers rules to find what numbers are multiples of small numbers. Here’s an example. Think of a three-digit number. Suppose its first digit is ‘a’, its second digit ‘b’, and its third digit ‘c’. So we’d write the number as ‘abc’, or, 100a + 10b + 1c. What’s this number equal to, modulo 9? Well, 100a modulo 9 has to be equal to whatever a modulo 9 is: (100 a) modulo 9 is (100) modulo 9 — that is, 1 — times (a) modulo 9. 10b modulo 9 is (10) modulo 9 — again, 1 — times (b) modulo 9. 1c modulo 9 is … well, (c) modulo 9. Add that all together and you have a + b + c modulo 9. If a + b + c is some multiple of 9, so must be 100a + 10b + 1c.

The rules about whether something’s divisible by 2 or 5 or 10 are easy to work with since 10 is a multiple of 2, and of 5, and for that matter of 10, so that 100a + 10b + 1c modulo 10 is just c modulo 10. You might want to let this settle. Then, if you like, practice by working out what an add-the-digits rule for multiples of 11 would be. (This is made a lot easier if you remember that 10 is equal to 11 – 1.) And if you want to show off some serious arithmetic skills, try working out an add-the-digits rule for finding whether something’s a multiple of 7. Then you’ll know why nobody has ever used that for any real work.

J C Duffy’s Lug Nuts plays on the equivalence people draw between intelligence and arithmetic ability. Also on the idea that brain size should have something particularly strong link to intelligence. Really anyone having trouble figuring out 15% of \$10 is psyching themselves out. They’re too much overwhelmed by the idea of percents being complicated to realize that it’s, well, ten times 15 cents.

## Reading the Comics, January 22, 2018: Breaking Workflow Edition

So I was travelling last week, and this threw nearly all my plans out of whack. We stayed at one of those hotels that’s good enough that its free Internet is garbage and they charge you by day for decent Internet. So naturally Comic Strip Master Command sent a flood of posts. I’m trying to keep up and we’ll see if I wrap up this past week in under three essays. And I am not helped, by the way, by GoComics.com rejiggering something on their server so that My Comics Page won’t load, and breaking their “Contact Us” page so that that won’t submit error reports. If someone around there can break in and turn one of their servers off and on again, I’d appreciate the help.

Hy Eisman’s Katzenjammer Kids for the 21st of January is a curiously-timed Tax Day joke. (Well, the Katzenjammer Kids lapsed into reruns a dozen years ago and there’s probably not much effort being put into selecting seasonally appropriate ones.) But it is about one of the oldest and still most important uses of mathematics, and one that never gets respect.

Morrie Turner’s Wee Pals rerun for the 21st gets Oliver the reputation for being a little computer because he’s good at arithmetic. There is something that amazes in a person who’s able to calculate like this without writing anything down or using a device to help.

Steve Kelley and Jeff Parker’s Dustin for the 22nd seems to be starting off with a story problem. It might be a logic problem rather than arithmetic. It’s hard to say from what’s given.

Mark Anderson’s Andertoons for the 22nd is the Mark Anderson’s Andertoons for the week. Well, for Monday, as I write this. It’s got your classic blackboard full of equations for the people in over their head. The equations look to me like gibberish. There’s a couple diagrams of aromatic organic compounds, which suggests some quantum-mechanics chemistry problem, if you want to suppose this could be narrowed down.

Greg Evans’s Luann Againn for the 22nd has Luann despair about ever understanding algebra without starting over from scratch and putting in excessively many hours of work. Sometimes it feels like that. My experience when lost in a subject has been that going back to the start often helps. It can be easier to see why a term or a concept or a process is introduced when you’ve seen it used some, and often getting one idea straight will cause others to fall into place. When that doesn’t work, trying a different book on the same topic — even one as well-worn as high school algebra — sometimes helps. Just a different writer, or a different perspective on what’s key, can be what’s needed. And sometimes it just does take time working at it all.

Richard Thompson’s Richard’s Poor Almanac rerun for the 22nd includes as part of a kit of William Shakespeare paper dolls the Typing Monkey. It’s that lovely, whimsical figure that might, in time, produce any written work you could imagine. I think I’d retired monkeys-at-typewriters as a thing to talk about, but I’m easily swayed by Thompson’s art and comic stylings so here it is.

Darrin Bell and Theron Heir’s Rudy Park for the 18th throws around a lot of percentages. It’s circling around the sabermetric-style idea that everything can be quantified, and measured, and that its changes can be tracked. In this case it’s comments on Star Trek: Discovery, but it could be anything. I’m inclined to believe that yeah, there’s an astounding variety of things that can be quantified and measured and tracked. But it’s also easy, especially when you haven’t got a good track record of knowing what is important to measure, to start tracking what amounts to random noise. (See any of my monthly statistics reviews, when I go looking into things like views-per-visitor-per-post-made or some other dubiously meaningful quantity.) So I’m inclined to side with Randy and his doubts that the Math Gods sanction this much data-mining.

## Reading the Comics, June 3, 2017: Feast Week Conclusion Edition

And now finally I can close out last week’s many mathematically-themed comic strips. I had hoped to post this Thursday, but the Why Stuff Can Orbit supplemental took up my writing energies and eventually timeslot. This also ends up being the first time I’ve had one of Joe Martin’s comic strips since the Houston Chronicle ended its comics pages and I admit I’m not sure how I’m going to work this. I’m also not perfectly sure what the comic strip means.

So Joe Martin’s Mister Boffo for the 1st of June seems to be about a disastrous mathematics exam with a kid bad enough he hasn’t even got numbers exactly to express the score. Also I’m not sure there is a way to link to the strip I mean exactly; the archives for Martin’s strips are not … organized the way I would have done. Well, they’re his business.

Greg Evans’s Luann Againn for the 1st reruns the strip from the 1st of June, 1989. It’s your standard resisting-the-word-problem joke. On first reading the strip I didn’t get what the problem was asking for, and supposed that the text had garbled the problem, if there were an original problem. That was my sloppiness is all; it’s a perfectly solvable question once you actually read it.

J C Duffy’s Lug Nuts for the 1st — another day that threatened to be a Reading the Comics post all on its own — is a straggler Pi Day joke. It’s just some Dadaist clowning about.

Doug Bratton’s Pop Culture Shock Therapy for the 1st is a wordplay joke that uses word problems as emblematic of mathematics. I’m okay with that; much of the mathematics that people actually want to do amounts to extracting from a situation the things that are relevant and forming an equation based on that. This is what a word problem is supposed to teach us to do.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 1st — maybe I should have done a Reading the Comics for that day alone — riffs on the idle speculation that God would be a mathematician. It does this by showing a God uninterested in two logical problems. The first is the question of whether there’s an odd perfect number. Perfect numbers are these things that haunt number theory. (Everything haunts number theory.) It starts with idly noticing what happens if you pick a number, find the numbers that divide into it, and add those up. For example, 4 can be divided by 1 and 2; those add to 3. 5 can only be divided by 1; that adds to 1. 6 can be divided by 1, 2, and 3; those add to 6. For a perfect number the divisors add up to the original number. Perfect numbers look rare; for a thousand years or so only four of them (6, 28, 496, and 8128) were known to exist.

All the perfect numbers we know of are even. More, they’re all numbers that can be written as the product $2^{p - 1} \cdot \left(2^p - 1\right)$ for certain prime numbers ‘p’. (They’re the ones for which $2^p - 1$ is itself a prime number.) What we don’t know, and haven’t got a hint about proving, is whether there are any odd prime numbers. We know some things about odd perfect numbers, if they exist, the most notable of them being that they’ve got to be incredibly huge numbers, much larger than a googol, the standard idea of an incredibly huge number. Presumably an omniscient God would be able to tell whether there were an odd perfect number, or at least would be able to care whether there were. (It’s also not known if there are infinitely many perfect numbers, by the way. This reminds us that number theory is pretty much nothing but a bunch of easy-to-state problems that we can’t solve.)

Some miscellaneous other things we know about an odd perfect number, other than whether any exist: if there are odd perfect numbers, they’re not divisible by 105. They’re equal to one more than a whole multiple of 12. They’re also 117 more than a whole multiple of 468, and they’re 81 more than a whole multiple of 324. They’ve got to have at least 101 prime factors, and there have to be at least ten distinct prime factors. There have to be at least twelve distinct prime factors if 3 isn’t a factor of the odd perfect number. If this seems like a screwy list of things to know about a thing we don’t even know exists, then welcome to number theory.

The beard question I believe is a reference to the logician’s paradox. This is the one postulating a village in which the village barber shaves all, but only, the people who do not shave themselves. Given that, who shaves the barber? It’s an old joke, but if you take it seriously you learn something about the limits of what a system of logic can tell you about itself.

Bud Blake’s Tiger rerun for the 2nd has Tiger’s arithmetic homework spill out into real life. This happens sometimes.

George Herriman’s Krazy Kat for the 10th of July, 1939 was rerun the 2nd of June. I’m not sure that it properly fits here, but the talk about Officer Pupp running at 60 miles per hour and Ignatz Mouse running forty and whether Pupp will catch Mouse sure reads like a word problem. Later strips in the sequence, including the ways that a tossed brick could hit someone who’d be running faster than it, did not change my mind about this. Plus I like Krazy Kat so I’ll take a flimsy excuse to feature it.

## Reading the Comics, March 4, 2017: Frazz, Christmas Trees, and Weddings Edition

It was another of those curious weeks when Comic Strip Master Command didn’t send quite enough comics my way. Among those they did send were a couple of strips in pairs. I can work with that.

Samson’s Dark Side Of The Horse for the 26th is the Roman Numerals joke for this essay. I apologize to Horace for being so late in writing about Roman Numerals but I did have to wait for Cecil Adams to publish first.

In Jef Mallett’s Frazz for the 26th Caulfield ponders what we know about Pythagoras. It’s hard to say much about the historical figure: he built a cult that sounds outright daft around himself. But it’s hard to say how much of their craziness was actually their craziness, how much was just that any ancient society had a lot of what seems nutty to us, and how much was jokes (or deliberate slander) directed against some weirdos. What does seem certain is that Pythagoras’s followers attributed many of their discoveries to him. And what’s certain is that the Pythagorean Theorem was known, at least a thing that could be used to measure things, long before Pythagoras was on the scene. I’m not sure if it was proved as a theorem or whether it was just known that making triangles with the right relative lengths meant you had a right triangle.

Greg Evans’s Luann Againn for the 28th of February — reprinting the strip from the same day in 1989 — uses a bit of arithmetic as generic homework. It’s an interesting change of pace that the mathematics homework is what keeps one from sleep. I don’t blame Luann or Puddles for not being very interested in this, though. Those sorts of complicated-fraction-manipulation problems, at least when I was in middle school, were always slogs of shuffling stuff around. They rarely got to anything we’d like to know.

Jef Mallett’s Frazz for the 1st of March is one of those little revelations that statistics can give one. Myself, I was always haunted by the line in Carl Sagan’s Cosmos about how, in the future, with the Sun ageing and (presumably) swelling in size and heat, the Earth would see one last perfect day. That there would most likely be quite fine days after that didn’t matter, and that different people might disagree on what made a day perfect didn’t matter. Setting out the idea of a “perfect day” and realizing there would someday be a last gave me chills. It still does.

Richard Thompson’s Poor Richard’s Almanac for the 1st and the 2nd of March have appeared here before. But I like the strip so I’ll reuse them too. They’re from the strip’s guide to types of Christmas trees. The Cubist Fur is described as “so asymmetrical it no longer inhabits Euclidean space”. Properly neither do we, but we can’t tell by eye the difference between our space and a Euclidean space. “Non-Euclidean” has picked up connotations of being so bizarre or even horrifying that we can’t hope to understand it. In practice, it means we have to go a little slower and think about, like, what would it look like if we drew a triangle on a ball instead of a sheet of paper. The Platonic Fir, in the 2nd of March strip, looks like a geometry diagram and I doubt that’s coincidental. It’s very hard to avoid thoughts of Platonic Ideals when one does any mathematics with a diagram. We know our drawings aren’t very good triangles or squares or circles especially. And three-dimensional shapes are worse, as see every ellipsoid ever done on a chalkboard. But we know what we mean by them. And then we can get into a good argument about what we mean by saying “this mathematical construct exists”.

Mark Litzler’s Joe Vanilla for the 3rd uses a chalkboard full of mathematics to represent the deep thinking behind a silly little thing. I can’t make any of the symbols out to mean anything specific, but I do like the way it looks. It’s quite well-done in looking like the shorthand that, especially, physicists would use while roughing out a problem. That there are subscripts with forms like “12” and “22” with a bar over them reinforces that. I would, knowing nothing else, expect this to represent some interaction between particles 1 and 2, and 2 with itself, and that the bar means some kind of complement. This doesn’t mean much to me, but with luck, it means enough to the scientist working it out that it could be turned into a coherent paper.

Bill Holbrook’s On The Fastrack is this week about the wedding of the accounting-minded Fi. And she’s having last-minute doubts, which is why the strip of the 3rd brings in irrational and anthropomorphized numerals. π gets called in to serve as emblematic of the irrational numbers. Can’t fault that. I think the only more famously irrational number is the square root of two, and π anthropomorphizes more easily. Well, you can draw an established character’s face onto π. The square root of 2 is, necessarily, at least two disconnected symbols and you don’t want to raise distracting questions about whether the root sign or the 2 gets the face.

That said, it’s a lot easier to prove that the square root of 2 is irrational. Even the Pythagoreans knew it, and a bright child can follow the proof. A really bright child could create a proof of it. To prove that π is irrational is not at all easy; it took mathematicians until the 19th century. And the best proof I know of the fact does it by a roundabout method. We prove that if a number (other than zero) is rational then the tangent of that number must be irrational, and vice-versa. And the tangent of π/4 is 1, so therefore π/4 must be irrational, so therefore π must be irrational. I know you’ll all trust me on that argument, but I wouldn’t want to sell it to a bright child.

Holbrook continues the thread on the 4th, extends the anthropomorphic-mathematics-stuff to call people variables. There’s ways that this is fair. We use a variable for a number whose value we don’t know or don’t care about. A “random variable” is one that could take on any of a set of values. We don’t know which one it does, in any particular case. But we do know — or we can find out — how likely each of the possible values is. We can use this to understand the behavior of systems even if we never actually know what any one of it does. You see how I’m going to defend this metaphor, then, especially if we allow that what people are likely or unlikely to do will depend on context and evolve in time.