## Reading the Comics, June 3, 2020: Subjective Opinions Edition

Thanks for being here for the last week before my All-2020 Mathematics A to Z starts. By the time this posts I should have decided on the A-topic, but I’m still up for B or C topics, if you’d be so kind as to suggest things.

Bob Weber Jr’s Slylock Fox for the 1st of June sees Reeky Rat busted for speeding on the grounds of his average speed. It does make the case that Reeky Rat must have travelled faster than 20 miles per hour at some point. There’s no information about when he did it, just the proof that there must have been some time when he drove faster than the speed limit. One can find loopholes in the reasoning, but, it’s a daily comic strip panel for kids. It would be unfair to demand things like proof there’s no shorter route from the diner and that the speed limit was 20 miles per hour the whole way.

Ted Shearer’s Quincy for the 1st originally ran the 7th of April, 1981. Quincy and his friend ponder this being the computer age, and whether they can let computers handle mathematics.

Jef Mallett’s Frazz for the 2nd has the characters talk about how mathematics offers answers that are just right or wrong. Something without “subjective grading”. It enjoys that reputation. But it’s not so, and that’s obvious when you imagine grading. How would you grade an answer that has the right approach, but makes a small careless error? Or how would you grade an approach that doesn’t work, but that plausibly could?

And how do you know that the approach wouldn’t work? Even in non-graded mathematics, we have subjectivity. Much of mathematics is a search for convincing arguments about some question. What we hope to be convinced of is that there is a sound logical argument making the same conclusions. Whether the argument is convincing is necessarily subjective.

Yes, in principle, we could create a full deductive argument. It will take forever to justify every step from some axiom or definition or rule of inference. And even then, how do we know a particular step is justified? It’s because we think we understand what the step does, and how it conforms to one (or more) rule. That’s again a judgement call.

(The grading of essays is also less subjective than you might think if you haven’t been a grader. The difference between an essay worth 83 points and one worth 85 points may be trivial, yes. But you will rarely see an essay that reads as an A-grade one day and a C-grade the next. This is not to say that essay grading is not subject to biases. Some of these are innocent, such as the way the grader’s mood will affect the grade. Or how the first several papers, or the last couple, will be less consistently graded than the ones done in the middle of the project. Some are pernicious, such as under-rating the work done by ethnic minority students. But these biases affect the way one would grade, say, the partial credit for an imperfectly done algebra problem too.)

Mark Anderson’s Andertoons for the 3rd is the Mark Anderson’s Andertoons for the week. I could also swear that I’ve featured it here before. I can’t find it, if I have discussed this strip before. I may not have. Wavehead’s observing the difference between zero as an additive identity and its role in multiplication.

Ryan Pagelow’s Buni for the 3rd fits into the anthropomorphic-numerals category of joke. It’s really more of a representation of the year as the four horsemen of the Apocalypse.

Dan Collins’s Looks Good on Paper for the 3rd has a cook grilling a “Möbius Strip Steak”. It’s a good joke for putting on a mathematics instructor’s door.

Doug Savage’s Savage Chickens for the 3rd has, as part of animal facts, the assertion that “llamas have basic math skills”. I don’t know of any specific research on llama mathematics skills. But animals do have mathematics skills. Often counting. Some amount of reasoning. Social animals often have an understanding of transitivity, as well, especially if the social groups have a pecking order.

And this wraps up half of the past week’s mathematically-themed comic strips. I hope to have the rest in a Reading the Comics post at this link in a few days. Thanks for reading.

## Reading the Comics, April 17, 2020: Creating Models Edition

And now let me close out a week ago, in the comics. It was a slow week and it finished on a bunch of casual mentions of mathematical topics.

Gary Larson’s The Far Side compilation “Hands Off My Bunsen Burner” features this panel creating a model of how to get rights out of wrongs. The material is a joke, but trying to find a transformation from one mathematical object to another is a reasonable enough occupation.

Ted Shearer’s Quincy rerun for the 15th is one in the lineage of strips about never using mathematics in later life. Quincy challenges us to think of a time a reporter asks the President how much is 34 times 587.

That’s an unpleasant multiplication to do. But I can figure some angles on it. 34 is just a bit over one-third of 100. 587 is just a bit under 600. So, 34 times 587 has to be tolerably near one-third of 100 times 600. So it should be something around 20,000. To get it more exact: 587 is 13 less than 600. So, 587 times one-third of a hundred will be 600 times one-third of a hundred minus 13 times one-third of a hundred. That’s one-third of 130, which is about 40. So the product has to be something close to 19,960. And the product has be some number which ends in an 8, what with 4 times 7 being 28. So the answer has to be one of 19,948, 19,958, or 19,968. And, indeed, it’s 19,958. I doubt I could do that so well during a press conference, I’ll admit. (If I wanted to be sure about that second digit, I’d have worked out: the tens unit in 34 times the ones in 587 is three times seven which is 21; the ones unit in 34 times the tens unit in 587 is four times eight which is 32; and the 4 times 7 being 28 gives me a 2 in the tens unit. So, 1 plus 2 plus 2 is 5, and there we go.)

Brian Anderson’s Dog Eat Doug for the 15th uses blackboards full of equations to represent deep thinking. I can’t make out what the symbols say. They look quite good, though, and seem to have the form of legitimate expressions.

Terri Liebenson’s The Pajama Diaries for the 17th imagines creating a model for the volume of a laundry pile. The problem may seem trivial, but it reflects an important kind of work. Many processes are about how something that’s always accumulating will be handled. There’s usually a hard limit to the rate at which whatever it is gets handled. And there’s usually very little reserve, in either capacity or time. This will cause, for example, a small increase in traffic in a neighborhood to produce great jams, or how a modest rain can overflow the whole city’s sewer systems. Or how a day of missing the laundry causes there to be a week’s backlog of dirty clothes.

And a little final extra comic strip. I don’t generally mention web comics here, except for those that have fallen in with a syndicator like GoComics.com. (This is not a value judgement against web comics. It’s that I have to stop reading sometime.) But Kat Swenski’s KatRaccoon Comics recently posted this nice sequence with a cat facing her worst fear: a calculus date.

And that’s my comics for a week ago. Later this week I’ll cover the past week’s handful of comics, in an essay at this link. Thanks for reading.

## Reading the Comics, March 7, 2020: Everybody Has Tests Edition

It was another pretty quiet week for mathematically-themed comic strips. Most of what did mention my subject just presented it as a subject giving them homework or quizzes or exams. But let’s look over what is here.

Morrie Turner’s Wee Pals rerun for the 3rd is an example of this, with one of the kids mourning his arithmetic grade. The strip previously ran the 3rd of March, 2015.

Hector D. Cantú and Carlos Castellanos’s Baldo for the 4th similarly has mathematics homework under review. And, you know, one of those mistakes that’s obvious if you do a quick “sanity check”, thinking over whether your answer could make sense.

Ted Shearer’s Quincy for the 5th is the most interesting strip of the week, since it suggests an actual answerable mathematics problem. How much does a professional basketball player earn per dribble? The answer requires a fair bit of thought, like, what do you mean by “a professional basketball player”? There’s many basketball leagues around the world; even if we limit the question to United States-and-Canada leagues, there’s a fair number of minor leagues. If we limit it to the National Basketball Association there’s the question of whether the salary is the minimum union contract guarantee, or the mean salary, or the median salary. It’s exciting to look at the salary of the highest-paid players, too, of course.

Working out the number of dribbles per year is also a fun estimation challenge. Even if we pick a representative player there’s no getting an exact count of how many dribbles they’ve made over a year, even if we just consider “dribbling during games” to be what’s paid for. (And any reasonable person would have to count all the dribbling done during warm-up and practice as part of what’s being paid for.) But someone could come up with an estimate of, for example, about how long a typical player has the ball for a game, and how much of that time is spent moving the ball or preparing for a free throw or other move that calls for dribbling. How long a dribble typically takes. How many games a player typically plays over the year. The estimate you get from this will never, ever, be exactly right. But it should be close enough to give an idea how much money a player earns in the time it takes to dribble the ball once. So occasionally the comics put forth a good story problem after all.

Quincy on the 7th is again worrying about his mathematics and spelling tests. It’s a cute coincidence that these are the subjects worried about in Wee Pals too.

Paul Gilligan’s Pooch Cafe for the 7th is part of a string of jokes about famous dogs. This one’s a riff on Albert Einstein, mentioned here because Albert Einstein has such strong mathematical associations.

And that’s all the week there was. I’ll be Reading the Comics for their mathematics content next week, too, and be glad to see you then. My guess is: some jokes about π.

## Reading the Comics, January 30, 2019: Interlude Edition

I think there are just barely enough comic strips from the past week to make three essays this time around. But one of them has to be a short group, only three comics. That’ll be for the next essay when I can group together all the strips that ran in February. One strip that I considered but decided not to write at length about was Ed Allison’s dadaist Unstrange Phenomena for the 28th. It mentions Roman Numerals and the idea of sneaking message in through them. But that’s not really mathematics. I usually enjoy the particular flavor of nonsense which Unstrange Phenomena uses; you might, too.

John McPherson’s Close to Home for the 29th uses an arithmetic problem as shorthand for an accomplished education. The problem is solvable. Of course, you say. It’s an equation with quadratic polynomial; it can hardly not be solved. Yes, fine. But McPherson could easily have thrown together numbers that implied x was complex-valued, or had radicals or some other strange condition. This is one that someone could do in their heads, at least once they practiced in mental arithmetic.

I feel reasonably confident McPherson was just having a giggle at the idea of putting knowledge tests into inappropriate venues. So I’ll save the full rant. But there is a long history of racist and eugenicist ideology that tried to prove certain peoples to be mentally incompetent. Making an arithmetic quiz prerequisite to something unrelated echoes that. I’d have asked McPherson to rework the joke to avoid that.

(I’d also want to rework the composition, since the booth, the swinging arm, and the skirted attendant with the clipboard don’t look like any tollbooth I know. But I don’t have an idea how to redo the layout so it’s more realistic. And it’s not as if that sort of realism would heighten the joke.)

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 29th riffs on the problem of squaring the circle. This is one of three classical problems of geometry. The lecturer describes it just fine: is it possible to make a square that’s got the same area as a given circle, using only straightedge and compass? There are shapes it’s easy to do this for, such as rectangles, parallelograms, triangles, and (why not?) this odd crescent-moon shaped figure called the lune. Circles defied all attempts. In the 19th century mathematicians found ways to represent the operations of classical geometry with algebra, and could use the tools of algebra to show squaring the circle was impossible. The squaring would be equivalent to finding a polynomial, with integer coefficients, that has $\sqrt{\pi}$ as a root. And we know from the way algebra works that this can’t be done. So squaring the circle can’t be done.

The lecturer’s hack, modifying the compass and straightedge, lets you in principle do whatever you want. The hack isn’t new either. Modifying the geometric tools changes what you can and can’t do. The Ancient Greeks recognized that adding some specialized tools would make the problem possible. But that falls outside the scope of the problem.

Which feeds to the secondary joke, of making the philosophers sad. Often philosophy problems test one’s intuition about an idea by setting out a problem, often with unpleasant choices. A common problem with students that I’m going ahead and guessing are engineers is then attacking the setup of the question, trying to show that the problem couldn’t actually happen. You know, as though there were ever a time significant numbers of people were being tied to trolley tracks. (By the way, that thing about silent movie villains tying women to railroad tracks? Only happened in comedies spoofing Victorian melodramas. It’s always been a parody.) Attacking the logic of a problem may make for good movie drama. But it makes for a lousy student and a worse class discussion.

Ted Shearer’s Quincy rerun for the 30th uses a bit of mathematics and logic talk. It circles the difference between the feeling one can have about the rational meaning of a situation and how the situation feels to someone. It seems like a jump that Quincy goes from being asked about logic to talking about arithmetic. Possibly Quincy’s understanding of logic doesn’t start from the sort of very abstract concept that makes arithmetic hard to get to, though.

There should be another Reading the Comics post this week. It should be here, when it appears. There should also be one on Sunday, as usual.

## Reading the Comics, September 14, 2018: I Already Forgot What I Said About Randolph Itch Edition

Yeah, so remember how like two weeks ago I noticed another Randolph Itch, 2 am repeat? And figured to retire the comic strip from my Reading the Comics routine? Well, then you’re better at this blog than I am. But this time I’ll retire it for sure, rather than waste text I wrote up already.

Tom Toles’s Randolph Itch, 2 am for the 13th is the Roman Numerals joke for the week. IV is a well-established way to write four, although on clock faces IIII is a quite common use. There’s not a really clear reason why this should be. I’m convinced that it’s mostly for reasons of symmetry. IIII comes nearer the length of VIII, across it on the clock face. The subtractive principle, where ‘IV’ means ‘one taken away from five’, wasn’t really a thing until the middle ages. But then neither were clocks like that.

Bill Rechin’s Crock for the 14th is a joke about being bad at arithmetic. And yeah, most instructors wouldn’t accept “a lot” as the answer to 125 times 140. But we can go from approximations to something more precise. The number’s got to be more than 10,000, for example. 125 is more than 100, and 140 is more than 100. So 125 times 140 has to be more than 100 times 100. And then I notice: 125 is a hundred plus a quarter-of-a-hundred. So, 125 times 140 is a hundred times 140 plus a quarter-of-a-hundred times 140. A hundred times 140 is easy: it’s 14,000. A quarter of that? … Is a quarter of 12,000 plus a quarter of 2,000. That’s 3,000 plus 500. So 125 times 140 has to be 14,000 plus 3,000 plus 500. 17,500. My calculator agrees, so I feel pretty good. If this all seems like an ad hoc process, well, it is. But it’s how I can do this in my head.

Yes, the comments at ComicsKingdom include a warning that “using this obscenity called new math he may never be right, but he will never be wrong either”. I mention this for fans of cranky old person comics commentary.

Ted Shearer’s Quincy for the 21st of July, 1979 was rerun the 14th. It expresses the then-common wish for a calculator, which held such promise for making mathematics easy. It does make some kinds of mathematics easy. It especially takes considerable tedium out of mathematics. And it opens up new things to discover. Especially if the calculator lets you put the last thing calculated into a formula. That makes it easy to play with all sorts of iterative processes. They let you find solutions to weird and complicated problems. Or explore beautiful fractals. Figure out what limits work like. Or just notice what’s neat about 3.302775638. They let you get into different things.

Daniel Shelton’s Ben for the 14th has Nicholas doing mathematics homework. And something that couldn’t just be any subject; arranging fractions by size is something worth learning. They do have the peculiar and hard-to-adjust-to property that making the denominator larger, without changing the numerator, makes the entire fraction represent a smaller number. I mean a number closer to zero. So I think sorting fractions a reasonable homework project. Cutting them out and pasting them down seems weird to me. But maybe there’s some benefit in making the project tactile like that.

My Reading the Comics posts make up the bulk of this blog by volume. They should all appear at this link. I really this time mean to retiree Randolph Itch, 2am as a tag, but please enjoy the strip’s appearances at this link. This and other appearances by Crock are at this link. Ted Shearer’s Quincy appears in essays on this link And other appearances by Ben should be at this link. Also I’m surprised to learn there are other essays. I would have bet Ben was a new tag this essay.

## Reading the Comics, July 14, 2018: County Fair Edition

The title doesn’t mean anything. My laptop’s random-draw of pictures pulled up one from the county fair last year is all. I’m just working too close to deadline to have a good one. Pet rabbit has surgery scheduled and we are hoping that turns out well for everyone involved.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 12th has the blackboard of mathematical symbols. Familiar old shorthand of conflating mathematics ability with genius, or at least intelligence. The blackboard isn’t particularly full of expressions, possibly because Caulfield and Rouillard’s art might not be able to render too much detail clearly. It’s also got a sort-of appearance of Einstein’s most famous equation. Although with perhaps an extra joke to it. Suppose we’re to take ‘E’ and ‘M’ and ‘C’ to mean what they do in Einstein’s use. Then $E - mc^2$ has to equal zero. And there are many things you can safely do with zero. Dividing by it, though, isn’t one. I shan’t guess whether Caulfield and Rouillard were being that sly, though.

Marty Links’s Emmy Lou rerun for the 13th tries to be a paradox. How can one like mathematics without liking figures? But arithmetic is just one part of mathematics. Surely the most-used part, if we go by real-world utility. But not everything. Arithmetic is often useful, yes. But you can do good work in (say) logic or knot theory or geometry with only a slight ability to add or subtract or multiply. There’s not enough emphasis put on that in early education. I suppose it reflects the reasonable feeling that people do need to be competent at arithmetic, which is useful. But it gives one a distorted view of what mathematics can be.

Mark Parisi’s Off The Markfor the 13th is the anthropomorphic numerals joke for the week. And it presents being multiplied by zero as a terrifying fate for other numbers. This seems to reflect the idea that being multiplied by zero is equivalent to being made into nothing. That it’s being killed. Zero enjoys this dual meaning, culturally, representing both a number and the concept of a thing that doesn’t exist and the concept of non-existence. If being turned from one number to another is a numeral murder, then a 2 sneaking in with a + sign would be at least as horrifying. But that joke wouldn’t work, and I know that too.

Olivia Jaimes’s Nancy for the 14th is another recreational-mathematics puzzle. I know nothing of Jaimes’s background but apparently it involves a keen interest in that kind of play that either makes someone love or hate mathematics. (Myself, I’m only slightly interested in these kinds of puzzles, most of the time.) This one — add one line to ‘fix’ the equation 5 + 5 + 5 + 5 = 555 — I hadn’t encountered before. Took some fuming to work it out. The obvious answer, of course, is to add a slash across the = sign so that it means “does not equal”.

But that answer’s dull. What mathematicians like are statements that are true and interesting. There are many things that 5 + 5 + 5 + 5 does not equal. Why single out 555 from that set? So negating the equals sign meets the specifications of the problem, slightly better than Nancy does herself. It doesn’t have the surprise of the answer Nancy’s teacher wants.

If you don’t get how to do it, highlight over the paragraph below for a hint.

There are actually three ways to add the stroke to make this equation true. The three ways are equivalent, though. Notice that the symbols on the board comprise strokes and curves and consider that the meaning of the symbol can be changed by altering the composition of those strokes and curves.

Ted Shearer’s Quincy for the 21st of May, 1979, and rerun the 14th is a joke about making mathematics problems relevant. And, yeah, I’ll give Mrs Glover credit for making problems that reflect stuff students know they’re going to have to deal with. Also that they may have already dealt with and so have some feeling for what plausible answers will be. It’s tough to find many problems like that which don’t repeat themselves too much. (“If your pants need a new patch every two months how many would you have in three years?”).

I do many Reading the Comics posts. Others like this one are here. For other essays that mention Mustard and Boloney, look to this link. I admit I’m surprised there’s anything there; I didn’t remember having written about it before For other discussions of Emmy Lou, try this link. For this and other times I’ve written about Off The Mark try this link. For Nancy content, try this link. And for other Quincy essays you can read this link. Thank you.

## 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, March 31, 2018: A Normal Week Edition

I have a couple loose rules about these Reading the Comics posts. At least one a week, whether there’s much to talk about or not. Not too many comics in one post, because that’s tiring to read and tiring to write. Trying to write up each day’s comics on the day mitigates that some, but not completely. So I tend to break up a week’s material if I can do, say, two posts of about seven strips each. This year, that’s been necessary; I’ve had a flood of comics on-topic or close enough for me to write about. This past week was a bizarre case. There really weren’t enough strips to break up the workload. It was, in short, a normal week, as strange as that is to see. I don’t know what I’m going to do Thursday. I might have to work.

Aaron McGruder’s Boondocks for the 25th of March is formally just a cameo mention of mathematics. There is some serious content to it. Whether someone likes to do a thing depends, to an extent, on whether they expect to like doing a thing. It seems likely to me that if a community encourages people to do mathematics, then it’ll have more people who do mathematics well. Mathematics does at least have the advantage that a lot of its fields can be turned into games. Or into things like games. Is one knot the same as another knot? You can test the laborious but inevitably correct way, trying to turn one into the other. Or you can find a polynomial that describes both knots and see if those two are the same polynomials. There’s fun to be had in this. I swear. And, of course, making arguments and finding flaws in other people’s arguments is a lot of mathematics. And good fun for anybody who likes that sort of thing. (This is a new tag for me.)

Ted Shearer’s Quincy for the 30th of January, 1979 and rerun the 26th names arithmetic as the homework Quincy’s most worried about. Or would like to put off the most. Harmless enough.

Mike Thompson’s Grand Avenue for the 26th is a student-resisting-the-problem joke. A variable like ‘x’ serves a couple of roles. One of them is the name for a number whose value we don’t explicitly know, but which we hope to work out. And that’s the ‘x’ seen here. The other role of ‘x’ is the name for a number whose value we don’t know and don’t particularly care about. Since those are different reasons to use ‘x’ maybe we ought to have different names for the concepts. But we don’t and there’s probably no separating them now.

Tony Cochran’s Agnes for the 27th grumbles that mathematics and clairvoyance are poorly taught. Well, everyone who loves mathematics grumbles that the subject is poorly taught. I don’t know what the clairvoyants think but I’ll bet the same.

Mark Pett’s Lucky Cow rerun for the 28th is about sudoku. As with any puzzle the challenge is having rules that are restrictive enough to be interesting. This is also true of any mathematical field, though. You want ideas that imply a lot of things are true, but that also imply enough interesting plausible things are not true.

Rick DeTorie’s One Big Happy rerun for the 30th has Ruthie working on a story problem. One with loose change, which seems to turn up a lot in story problems. I never think of antes for some reason.

Stephen Beals’s Adult Children for the 31st depicts mathematics as the stuff of nightmares. (Although it’s not clear to me this is meant to recount a nightmare. Reads like it, anyway.) Calculus, too, which is an interesting choice. Calculus seems to be a breaking point for many people. A lot of people even who were good at algebra or trigonometry find all this talk about differentials and integrals and limits won’t cohere into understanding. Isaac Asimov wrote about this several times, and the sad realization that for as much as he loved mathematics there were big important parts of it that he could not comprehend.

I’m curious why calculus should be such a discontinuity, but the reasons are probably straightforward. It’s a field where you’re less interested in doing things to numbers and more interested in doing things to functions. Or to curves that a function might represent. It’s a field where information about a whole region is important, rather than information about a single point. It’s a field where you can test your intuitive feeling for, say, a limit by calculating a couple of values, but for which those calculations don’t give the right answer. Or at least can’t be guaranteed to be right. I don’t know if the choice of what to represent mathematics was arbitrary. But it was a good choice certainly. (This is another newly-tagged strip.)

## Reading the Comics, February 10, 2018: I Meant To Post This Thursday Edition

Ah, yes, so, in the midst of feeling all proud that I’d gotten my Reading the Comics workflow improved, I went out to do my afternoon chores without posting the essay. I’m embarrassed. But it really only affects me looking at the WordPress Insights page. It publishes this neat little calendar-style grid that highlights the days when someone’s posted and this breaks up the columns. This can only unnerve me. I deserve it.

Tom Thaves’s Frank and Ernest for the 8th of February is about the struggle to understand zero. As often happens, the joke has a lot of truth to it. Zero bundles together several ideas, overlapping but not precisely equal. And part of that is the idea of “nothing”. Which is a subtly elusive concept: to talk about the properties of a thing that does not exist is hard. As adults it’s easy to not notice this anymore. Part’s likely because mastering a concept makes one forget what it took to understand. Part is likely because if you don’t have to ponder whether the “zero” that’s “one less than one” is the same as the “zero” that denotes “what separates the count of thousands from the count of tens in the numeral 2,038” you might not, and just assume you could explain the difference or similarity to someone who has no idea.

John Zakour and Scott Roberts’s Maria’s Day for the 8th has maria and another girl bonding over their hatred of mathematics. Well, at least they’re getting something out of it. The date in the strip leads me to realize this is probably a rerun. I’m not sure just when it’s from.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th proposes a prank based on mathematical use of the word “arbitrarily”. This is a word that appears a lot in analysis, and the strip makes me realize I’m not sure I can give a precise definition. An “arbitrarily large number”, for example, would be any number that’s large enough. But this also makes me realize I’m not sure precisely what joke Weinersmith is going for. I suppose that if someone were to select an arbitrarily large number they might pick 53, or a hundred, or million billion trillion. I suppose Weinersmith’s point is that in ordinary speech an arbitrarily made choice is one selection from all the possible alternatives. In mathematical speech an arbitrarily made choice reflects every possible choice. To speak of an arbitrarily large number is to say that whatever selection is made, we can go on to show this interesting stuff is true. We’d typically like to prove the most generically true thing possible. But picking a single example can be easier to prove. It can certainly be easier to visualize. 53 is probably easier to imagine than “every number 52 or larger”, for example.

Ted Shearer’s Quincy for the 16th of December, 1978 was rerun the 9th of February. It just shows Quincy at work on his mathematics homework, and considering dedicating it to his grandmother. Mathematics books have dedications, just as any other book does. I’m not aware of dedications of proofs or other shorter mathematics works, but there’s likely some. There’s often a note of thanks, usually given to people who’ve made the paper’s writers think harder about the subjects. But I don’t think there’s any reason a paper wouldn’t thank someone who provided “mere” emotional support. I just don’t have examples offhand.

Jef Mallet’s Frazz for the 9th looks like one of those creative-teaching exercises I sometimes see in Mathematics Education Twitter: the teacher gives answers and the students come up with story problems to match. That’s not a bad project. I’m not sure how to grade it, but I haven’t done anything that creative when I’ve taught. I’m sorry I haven’t got more to say about it since the idea seems fun.

Gordon Bess’s Redeye for the 30th of September, 1971 was rerun the 10th. It’s a bit of extremely long division and I don’t blame Pokey for giving up on that problem. Starting from 5,967,342 divided by 973 I’d say, well, that’s about six million divided by a thousand, so the answer should be near six thousand. I don’t think the last digits of 2 and 3 suggest anything about what the final digit should be, if this divides evenly. So the only guidance I have is that my answer ought to be around six thousand and then we have to go into actually working. It turns out that 973 doesn’t go into 5,967,342 a whole number of times, so I sympathize more with Pokey. The answer is a little more than 6,132.9311.

## Reading the Comics, November 11, 2017: Pictured Comics Edition

And now the other half of last week’s comic strips. It was unusually rich in comics that come from Comics Kingdom or Creators.com, which have limited windows of access and therefore make me feel confident I should include the strips so my comments make any sense.

Rick Kirkman and Jerry Scott’s Baby Blues for the 9th mentions mathematics homework as a resolutely rage-inducing topic. It’s mathematics homework, obviously, or else it wouldn’t be mentioned around here. And even more specifically it’s Common Core mathematics homework. So it always is with attempts to teach subjects better. Especially mathematics, given how little confidence people have in their own mastery. I can’t blame parents for supposing any change to be just malice.

Bill Amend’s FoxTrot Classics for the 9th is about random numbers. As Jason says, it is hard to generate random numbers. Random numbers are a resource. Having a good source of them makes a lot of computation work. But they’re hard to make. It seems to be a contradiction to create random numbers by an algorithm. There’s reasons we accept pseudorandom numbers, or find quasirandom numbers. This strip originally ran the 16th of November, 2006.

Chris Browne’s Hagar the Horrible for the 10th is about the numerous. There’s different kinds of limits. There’s the greatest number of things we can count in an instant. There’s a limit to how long a string of digits or symbols we can remember. There’s the biggest number of things we can visualize. And “visualize” is a slippery concept. I think I have a pretty good idea what we mean when we say “a thousand” of something. I could calculate how long it took me to do something a thousand times, or to write a thousand of something. I know that it was at about a thousand words that, last A To Z sequence, I got to feeling I should wrap up any particular essay. But did I see any particular difference between word 999 and word 1,000? No; what I really knew was “about enough paragraphs” and maybe “fills just over two screens in my text editor”. So do I know what a thousand is? Anyway, we all have our limits, acknowledge them or not.

Henry Scarpelli and Craig Boldman’s Archie rerun for the 17th is about Moose’s struggle with mathematics. Just writing “more or less” doesn’t fix an erroneous answer, true. But error margins, and estimates of where an answer should be, can be good mathematics. (Part of the Common Core that many parents struggle with is making the estimate of an answer the first step, and a refined answer later. Based on what I see crossing social media, this really offends former engineering majors who miss the value in having an expected approximate answer.) It’s part of how we define limits, and derivatives, and integrals, and all of calculus. But it’s in a more precise way than Moose tries to do.

Ted Shearer’s Quincy for the 18th of September, 1978 is a story-problem joke. Some of these aren’t complicated strips.

## Reading the Comics, July 1, 2017: Deluge Edition, Part 2

Last week started off going like Gangbusters, a phrase I think that’s too old-fashioned for my father to say but that I’ve picked up because I like listening to old-time radio and, you know, Gangbusters really does get going like that. Give it a try sometime, if you’re open to that old-fashioned sort of narrative style and blatant FBI agitprop. You might want to turn the volume down a little before you do. It slowed down the second half of the week, which is mostly fine as I’d had other things taking up my time. Let me finish off last week and hope there’s a good set of comics to review for next Sunday and maybe Tuesday.

Ted Shearer’s Quincy for the 4th of May, 1978 was rerun the 28th of June. It’s got the form of your student-resisting-the-word-problem joke. And mixes in a bit of percentages which is all the excuse I need to include it here. That and how Shearer uses halftone screening. It’s also a useful reminder of how many of our economic problems could be solved quickly if poor people got more money.

Olivia Walch’s Imogen Quest for the 28th features Gottfried Leibniz — missing his birthday by three days, incidentally — and speaks of the priority dispute about the invention of calculus. I’m not sure there is much serious questioning anymore about Leibniz’s contributions to mathematics. I think they might be even more strongly appreciated these days than they ever used to be, as people learn more about his work in computing machines and the attempt to automate calculation.

Mark Anderson’s Andertoons for the 28th is our soothing, familiar Andertoons for this essay. I remember in learning about equivalent forms of fractions wondering why anyone cared about reducing them. If two things have the same meaning, why do we need to go further? There are a couple answers. One is that it’s easier on us to understand a quantity if it’s a shorter, more familiar form. $\frac{3}{4}$ has a meaning that $\frac{1131}{1508}$ just does not. And another is that we often want to know whether two things are equivalent, or close. Is \frac{1147}{1517} more or less than $\frac{1131}{1508}$? Good luck eyeballing that.

And we learn, later on, that a lot of mathematics is about finding different ways to write the same thing. Each way has its uses. Sometimes a slightly more complicated way to write a thing makes proving something easier. There’s about two solids months of Real Analysis, for example, where you keep on writing that $x_{n} - x_{m} \equiv x_{n} - x + x - x_{m}$ and this “adding zero” turns out to make proofs possible. Even easy.

Mark Tatulli’s Heart of the City remains on my watch-with-caution list as the Math Camp story continues. But the strip from the 28th tickles me with the idea of crossing mathematics camp with Pinocchio‘s Pleasure Island. I’m imagining something where Heart starts laughing at something and ends up turning into something from Donald Duck’s Mathmagic land.

Dave Blazek’s Loose Parts for the 28th is your traditional blackboard-full-of-symbols joke. I’m amused.

Tony Rubino and Gary Markstein’s Daddy’s Home for the 1st of July is your traditional “mathematics is something hard” joke. I have the feeling it’s a rerun, but I lack the emotional investment in whether it is a rerun to check. The joke’s comfortable and familiar as it is, anyway.

## Reading the Comics, June 24, 2017: Saturday Morning Breakfast Cereal Edition

Somehow this is not the title of every Reading The Comics review! But it is for this post and we’ll explore why below.

Dave Coverly’s Speed Bump for the 18th is not exactly an anthropomorphic-numerals joke. It is about making symbols manifest in the real world, at least. The greater-than and less-than signs as we know them were created by the English mathematician Thomas Harriot, and introduced to the world in his posthumous Artis Analyticae Praxis (1631). He also had an idea of putting a . between the numerals of an expression and the letters multiplied by them, for example, “4.x” to mean four times x. We mostly do without that now, taking multiplication as assumed if two meaningful quantities are put next to one another. But we will use, now, a vertically-centered dot to separate terms multiplied together when that helps our organization. The equals sign we trace to the 16th century mathematician Robert Recorde, whose 1557 Whetsone of Witte uses long but recognizable equals signs. The = sign went into hibernation after that, though, until the 17th century and it took some time to quite get well-used. So it often is with symbols.

Ted Shearer’s Quincy for the 25th of April, 1978 and rerun the 19th of June, starts from the history of zero. It’s worth noting there are a couple of threads woven together in the concept of zero. One is the idea of “nothing”, which we’ve had just forever. I mean, the idea that there isn’t something to work with. Another is the idea of the … well, the additive identity, there being some number that’s one less than one and two less than two. That you can add to anything without changing the thing. And then there’s symbols. There’s the placeholder for “there are no examples of this quantity here”. There’s the denotation of … well, the additive identity. All these things are zeroes, and if you listen closely, they are not quite the same thing. Which is not weird. Most words mean a collection of several concepts. We’re lucky the concepts we mean by “zero” are so compatible in meaning. Think of the poor person trying to understand the word “bear”, or “cleave”.

John Deering’s Strange Brew for the 19th is a “New Math” joke, fittingly done with cavemen. Well, numerals were new things once. Amusing to me is that — while I’m not an expert — in quite a few cultures the symbol for “one” was pretty much the same thing, a single slash mark. It’s hard not to suppose that numbers started out with simple tallies, and the first thing to tally might get dressed up a bit with serifs or such but is, at heart, the same thing you’d get jabbing a sharp thing into a soft rock.

Guy Gilchrist’s Today’s Dogg for the 19th I’m sure is a rerun and I think I’ve featured it here before. So be it. It’s silly symbol-play and dog arithmetic. It’s a comic strip about how dogs are cute; embrace it or skip it.

Zach Weinersmith’s Saturday Morning Breakfast Cereal is properly speaking reruns when it appears on GoComics.com. For whatever reason Weinersmith ran a patch of mathematics strips there this past week. So let me bundle all that up. On the 19th he did a joke mathematicians get a lot, about how the only small talk anyone has about mathematics is how they hated mathematics. I’m not sure mathematicians have it any better than any other teachers, though. Have you ever known someone to say, “My high school gym class gave me a greater appreciation of the world”? Or talk about how grade school history opened their eyes to the wonders of the subject? It’s a sad thing. But there are a lot of things keeping teachers from making students feel joy in their subjects.

For the 21st Weinersmith makes a statisticians joke. I can wrangle some actual mathematics out of an otherwise correctly-formed joke. How do we ever know that something is true? Well, we gather evidence. But how do we know the evidence is relevant? Even if the evidence is relevant, how do we know we’ve interpreted it correctly? Even if we have interpreted it correctly, how do we know that it shows what we want to know? Statisticians become very familiar with hypothesis testing, which amounts to the question, “does this evidence indicate that some condition is implausibly unlikely”? And they can do great work with that. But “implausibly unlikely” is not the same thing as “false”. A person knowledgeable enough and honest turns out to have few things that can be said for certain.

The June 23rd strip I’ve seen go around Mathematics Twitter several times, as see above tweet, about the ways in which mathematical literacy would destroy modern society. It’s a cute and flattering portrait of mathematics’ power, probably why mathematicians like passing it back and forth. But … well, how would “logic” keep people from being fooled by scams? What makes a scam work is that the premise seems logical. And real-world problems — as opposed to logic-class problems — are rarely completely resolvable by deductive logic. There have to be the assumptions, the logical gaps, and the room for humbuggery that allow hoaxes and scams to slip through. And does anyone need a logic class to not “buy products that do nothing”? And what is “nothing”? I have more keychains than I have keys to chain, even if we allow for emergencies and reasonable unexpected extra needs. This doesn’t stop my buying keychains as souvenirs. Does a Penn Central-logo keychain “do nothing” merely because it sits on the windowsill rather than hold any sort of key? If so, was my love foolish to buy it as a present? Granted that buying a lottery ticket is a foolish use of money; is my life any worse for buying that than, say, a peanut butter cup that I won’t remember having eaten a week afterwards? As for credit cards — It’s not clear to me that people max out their credit cards because they don’t understand they will have to pay it back with interest. My experience has been people max out their credit cards because they have things they must pay for and no alternative but going further into debt. That people need more money is a problem of society, yes, but it’s not clear to me that a failure to understand differential equations is at the heart of it. (Also, really, differential equations are overkill to understand credit card debt. A calculator with a repeat-the-last-operation feature and ten minutes to play is enough.)

## Reading the Comics, April 24, 2017: Reruns Edition

I went a little wild explaining the first of last week’s mathematically-themed comic strips. So let me split the week between the strips that I know to have been reruns and the ones I’m not so sure were.

Bill Amend’s FoxTrot for the 23rd — not a rerun; the strip is still new on Sundays — is a probability question. And a joke about story problems with relevance. Anyway, the question uses the binomial distribution. I know that because the question is about doing a bunch of things, homework questions, each of which can turn out one of two ways, right or wrong. It’s supposed to be equally likely to get the question right or wrong. It’s a little tedious but not hard to work out the chance of getting exactly six problems right, or exactly seven, or exactly eight, or so on. To work out the chance of getting six or more questions right — the problem given — there’s two ways to go about it.

One is the conceptually easy but tedious way. Work out the chance of getting exactly six questions right. Work out the chance of getting exactly seven questions right. Exactly eight questions. Exactly nine. All ten. Add these chances up. You’ll get to a number slightly below 0.377. That is, Mary Lou would have just under a 37.7 percent chance of passing. The answer’s right and it’s easy to understand how it’s right. The only drawback is it’s a lot of calculating to get there.

So here’s the conceptually harder but faster way. It works because the problem says Mary Lou is as likely to get a problem wrong as right. So she’s as likely to get exactly ten questions right as exactly ten wrong. And as likely to get at least nine questions right as at least nine wrong. To get at least eight questions right as at least eight wrong. You see where this is going: she’s as likely to get at least six right as to get at least six wrong.

There’s exactly three possibilities for a ten-question assignment like this. She can get four or fewer questions right (six or more wrong). She can get exactly five questions right. She can get six or more questions right. The chance of the first case and the chance of the last have to be the same.

So, take 1 — the chance that one of the three possibilities will happen — and subtract the chance she gets exactly five problems right, which is a touch over 24.6 percent. So there’s just under a 75.4 percent chance she does not get exactly five questions right. It’s equally likely to be four or fewer, or six or more. Just-under-75.4 divided by two is just under 37.7 percent, which is the chance she’ll pass as the problem’s given. It’s trickier to see why that’s right, but it’s a lot less calculating to do. That’s a common trade-off.

Ruben Bolling’s Super-Fun-Pax Comix rerun for the 23rd is an aptly titled installment of A Million Monkeys At A Million Typewriters. It reminds me that I don’t remember if I’d retired the monkeys-at-typewriters motif from Reading the Comics collections. If I haven’t I probably should, at least after making a proper essay explaining what the monkeys-at-typewriters thing is all about.

Ted Shearer’s Quincy from the 28th of February, 1978 reveals to me that pocket calculators were a thing much earlier than I realized. Well, I was too young to be allowed near stuff like that in 1978. I don’t think my parents got their first credit-card-sized, solar-powered calculator that kind of worked for another couple years after that. Kids, ask about them. They looked like good ideas, but you could use them for maybe five minutes before the things came apart. Your cell phone is so much better.

Bil Watterson’s Calvin and Hobbes rerun for the 24th can be classed as a resisting-the-word-problem joke. It’s so not about that, but who am I to slow you down from reading a Calvin and Hobbes story?

Garry Trudeau’s Doonesbury rerun for the 24th started a story about high school kids and their bad geography skills. I rate it as qualifying for inclusion here because it’s a mathematics teacher deciding to include more geography in his course. I was amused by the week’s jokes anyway. There’s no hint given what mathematics Gil teaches, but given the links between geometry, navigation, and geography there is surely something that could be relevant. It might not help with geographic points like which states are in New England and where they are, though.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 24th is built on a plot point from Carl Sagan’s science fiction novel Contact. In it, a particular “message” is found in the digits of π. (By “message” I mean a string of digits that are interesting to us. I’m not sure that you can properly call something a message if it hasn’t got any sender and if there’s not obviously some intended receiver.) In the book this is an astounding thing because the message can’t be; any reasonable explanation for how it should be there is impossible. But short “messages” are going to turn up in π also, as per the comic strips.

I assume the peer review would correct the cartoon mathematicians’ unfortunate spelling of understanding.

## Reading the Comics, January 7, 2016: Just Before GoComics Breaks Everything Edition

Most of the comics I review here are printed on GoComics.com. Well, most of the comics I read online are from there. But even so I think they have more comic strips that mention mathematical themes. Anyway, they’re unleashing a complete web site redesign on Monday. I don’t know just what the final version will look like. I know that the beta versions included the incredibly useful, that is to say dumb, feature where if a particular comic you do read doesn’t have an update for the day — and many of them don’t, as they’re weekly or three-times-a-week or so — then it’ll show some other comic in its place. I mean, the idea of encouraging people to find new comics is a good one. To some extent that’s what I do here. But the beta made no distinction between “comic you don’t read because you never heard of Microcosm” and “comic you don’t read because glancing at it makes your eyes bleed”. And on an idiosyncratic note, I read a lot of comics. I don’t need to see Dude and Dude reruns in fourteen spots on my daily comics page, even if I didn’t mind it to start.

Anyway. I am hoping, desperately hoping, that with the new site all my old links to comics are going to keep working. If they don’t then I suppose I’m just ruined. We’ll see. My suggestion is if you’re at all curious about the comics you read them today (Sunday) just to be safe.

Ashleigh Brilliant’s Pot-Shots is a curious little strip I never knew of until GoComics picked it up a few years ago. Its format is compellingly simple: a little illustration alongside a wry, often despairing, caption. I love it, but I also understand why was the subject of endless queries to the Detroit Free Press (Or Whatever) about why was this thing taking up newspaper space. The strip rerun the 31st of December is a typical example of the strip and amuses me at least. And it uses arithmetic as the way to communicate reasoning, both good and bad. Brilliant’s joke does address something that logicians have to face, too. Whether an argument is logically valid depends entirely on its structure. If the form is correct the reasoning may be excellent. But to be sound an argument has to be correct and must also have its assumptions be true. We can separate whether an argument is right from whether it could ever possibly be right. If you don’t see the value in that, you have never participated in an online debate about where James T Kirk was born and whether Spock was the first Vulcan in Star Fleet.

Thom Bluemel’s Birdbrains for the 2nd of January, 2017, is a loaded-dice joke. Is this truly mathematics? Statistics, at least? Close enough for the start of the year, I suppose. Working out whether a die is loaded is one of the things any gambler would like to know, and that mathematicians might be called upon to identify or exploit. (I had a grandmother unshakably convinced that I would have some natural ability to beat the Atlantic City casinos if she could only sneak the underaged me in. I doubt I could do anything of value there besides see the stage magic show.)

Jack Pullan’s Boomerangs rerun for the 2nd is built on the one bit of statistical mechanics that everybody knows, that something or other about entropy always increasing. It’s not a quantum mechanics rule, but it’s a natural confusion. Quantum mechanics has the reputation as the source of all the most solid, irrefutable laws of the universe’s working. Statistical mechanics and thermodynamics have this musty odor of 19th-century steam engines, no matter how much there is to learn from there. Anyway, the collapse of systems into disorder is not an irrevocable thing. It takes only energy or luck to overcome disorderliness. And in many cases we can substitute time for luck.

Scott Hilburn’s The Argyle Sweater for the 3rd is the anthropomorphic-geometry-figure joke that’s I’ve been waiting for. I had thought Hilburn did this all the time, although a quick review of Reading the Comics posts suggests he’s been more about anthropomorphic numerals the past year. This is why I log even the boring strips: you never know when I’ll need to check the last time Scott Hilburn used “acute” to mean “cute” in reference to triangles.

Mike Thompson’s Grand Avenue uses some arithmetic as the visual cue for “any old kind of schoolwork, really”. Steve Breen’s name seems to have gone entirely from the comic strip. On Usenet group rec.arts.comics.strips Brian Henke found that Breen’s name hasn’t actually been on the comic strip since May, and D D Degg found a July 2014 interview indicating Thompson had mostly taken the strip over from originator Breen.

Mark Anderson’s Andertoons for the 5th is another name-drop that doesn’t have any real mathematics content. But come on, we’re talking Andertoons here. If I skipped it the world might end or something untoward like that.

Ted Shearer’s Quincy for the 14th of November, 1977, doesn’t have any mathematical content really. Just a mention. But I need some kind of visual appeal for this essay and Shearer is usually good for that.

Corey Pandolph, Phil Frank, and Joe Troise’s The Elderberries rerun for the 7th is also a very marginal mention. But, what the heck, it’s got some of your standard wordplay about angles and it’ll get this week’s essay that much closer to 800 words.