Reading the Comics, March 28, 2020: Closing A Week Edition


I know; I’m more than a week behind the original publication of these strips. The Playful Math Education Blog Carnival took a lot of what attention I have these days. I’ll get caught up again soon enough. Comic Strip Master Command tried to help me, by having the close of a week ago being pretty small mathematics mentions, too. For example:

Thaves’s Frank and Ernest for the 26th is the anthropomorphic numerals joke for the week. Also anthropomorphic letters, for a bonus.

Craig Boldman and Henry Scarpelli’s Archie for the 27th has Moose struggling in mathematics this term. This is an interesting casual mention; the joke, of Moose using three words to describe a thing he said he could in two, would not fit sharply for anything but mathematics. Or, possibly, a measuring class, but there’s no high school even in fiction that has a class in measuring.

Bud Blake’s Vintage Tiger for the 27th has Tiger and Hugo struggling to find adjective forms for numbers. We can giggle at Hugo struggling for “quadruple” and going for something that makes more sense. We all top out somewhere, though, probably around quintuple or sextuple. I have never known anyone who claimed to know what the word would be for anything past decuple, and even staring at the dictionary page for “decuple” I don’t feel confident in it.

Hilary Price’s Rhymes With Orange for the 28th uses a blackboard full of calculations as shorthand for real insight into science. From context they’re likely working on some physics problem and it’s quite hard to do that without mathematics, must agree.

Ham’s Life On Earth for the 28th uses E = mc^2 as a milestone in a child’s development.

John Deering’s Strange Brew for the 28th name-drops slide rules, which, yeah, have mostly historical or symbolic importance these days. There might be some niche where they’re particularly useful (besides teaching logarithms), but I don’t know of it.


And what of the strips from last week? I’ll discuss them in an essay at this link, soon, I hope. Take care, please.

Reading the Comics, March 25, 2020: Regular Old Mathematics Mentions Edition


I haven’t forgotten about the comic strips. It happens that last week’s were mostly quite casual mentions, strips that don’t open themselves up to deep discussions. I write this before I see what I actually have to write about the strips. But here’s the first half of the past week’s. I’ll catch up on things soon.

Bill Amend’s FoxTrot for the 22nd, a new strip, has Jason and Marcus using arithmetic problems to signal pitches. At heart, the signals between a pitcher and catcher are just an index. They’re numbers because that’s an easy thing to signal given that one only has fingers and that they should be visually concealed. I would worry, in a pattern as complicated as these two would work out, about error correction. If one signal is mis-read — as will happen — how do they recognize it, and how do they fix it? This may seem like a lot of work to put to a trivial problem, but to conceal a message is important, whatever the message is.

Marcus, signalling a pitch: 'Two ... plus ... two ... minus .. one ... point ... three ... ' Jason, to Peter: 'If teams want to steal our signals, they're welcome to try.' Marcus: 'Can I just use a slash for 'divided by'?'.
Bill Amend’s FoxTrot for the 22nd of March, 2020. Essays mentioning either the new-run, Sunday, strips or the rerun, weekday, FoxTrot strips are gathered at this link.

Jerry Scott and Jim Borgman’s Zits for the 23rd has Jeremy preparing for a calculus test. Could be any subject.

James Beutel’s Banana Triangle for the 23rd has a character trying to convince himself os his intelligence. And doing so by muttering mathematics terms, mostly geometry. It’s a common shorthand to represent deep thinking.

Tom Batiuk’s Funky Winkerbean Vintage strip for the 24th, originally run the 13th of May, 1974, is wordplay about acute triangles.

Hector D Cantú and Carlos Castellanos’s Baldo for the 25th has Gracie work out a visual joke about plus signs. Roger Price, name-checked here, is renowned for the comic feature Droodles, extremely minimalist comic panels. He also, along with Get Smart’s Leonard Stern, created Mad Libs.

Man wrapped in flame, standing before God: 'Oh, come on! Grant me that I was within an order of magnitude of believing in the correct number of deities!'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 25th of March, 2020. It is quite common for me to write about this strip. You can see me explaining Saturday Morning Breakfast Cereal at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 25th is a joke about orders of magnitude. The order of magnitude is, roughly, how big the number is. Often the first step of a physics problem is to try to get a calculation that’s of the right order of magnitude. Or at least close to the order of magnitude. This may seem pretty lax. If we want to find out something with value, say, 231, it seems weird to claim victory that our model says “it will be a three-digit number”. But getting the size of the number right is a first step. For many problems, particularly in cosmology or astrophysics, we’re intersted in things whose functioning is obscure. And relies on quantities we can measure very poorly. This is why we can see getting the order magnitude about right as an accomplishment.


There’s another half-dozen strips from last week that at least mention mathematics. I’ll at least mention them soon, in an essay at this link. Thank you.

Reading the Comics, March 21, 2020: Pragmatic Calculations Edition


There were a handful of other comic strips last week. If they have a common theme (and I’ll try to drag one out) it’s that they circle around pragmatism. Not just using mathematics in the real world but the fussy stuff of what you can calculate and what you can use a calculation for.

And, again, I am hosting the Playful Math Education Blog Carnival this month. If you’ve run across any online tool that teaches mathematics, or highlights some delightful feature of mathematics? Please, let me know about it here, and let me know what of your own projects I should feature with it. The goal is to share things about mathematics that helped you understand more of it. Even if you think it’s a slight thing (“who cares if you can tell whether a number’s divisible by 11 by counting the digits right?”) don’t worry. Slight things count. Speaking of which …

Jef Mallett’s Frazz for the 20th has a kid ask about one of those add-the-digits divisibility tests. What happens if the number is too big to add up all the digits? In some sense, the question is meaningless. We can imagine finding the sum of digits no matter how many digits there are. At least if there are finitely many digits.

But there is a serious mathematical question here. We accept the existence of numbers so big no human being could ever know their precise value. At least, we accept they exist in the same way that “4” exists. If a computation can’t actually be finished, then, does it actually mean anything? And if we can’t figure a way to shorten the calculation, the way we can usually turn the infinitely-long sum of a series into a neat little formula?

Kid: 'A number is divisible by 3 if the sum of its digits is divisible by 3. But what if the number is so big there's too many digits to add up easily?' Frazz: 'If it's that big, the 1 or 2 left over isn't going to matter much.' Kid: 'Why don't they teach THAT kind of math more in school?' Frazz: 'I guess there's only jobs for so many songwriters, cartoonists, and janitors.'
Jef Mallett’s Frazz for the 20th of March, 2020. Essays featuring some topic raised by Frazz should be gathered at this link.

This gets into some cutting-edge mathematics. For calculations, some. But also, importantly, for proofs. A proof is, really, a convincing argument that something is true. The ideal of this is a completely filled-out string of logical deductions. These will take a long while. But, as long as it takes finitely many steps to complete, we normally accept the proof as done. We can imagine proofs that take more steps to complete than could possibly be thought out, or checked, or confirmed. We, living in the days after Gödel, are aware of the idea that there are statements which are true but unprovable. This is not that. Gödel’s Incompleteness Theorems tell us about statements that a deductive system can’t address. This is different. This is things that could be proven true (or false), if only the universe were more vast than it is.

There are logicians who work on the problem of what too-long-for-the-universe proofs can mean. Or even what infinitely long proofs can mean, if we allow those. And how they challenge our ideas of what “proof” and “knowledge” and “truth” are. I am not among these people, though, and can’t tell you what interesting results they have concluded. I just want to let you know the kid in Frazz is asking a question you can get a spot in a mathematics or philosophy department pondering. I mean so far as it’s possible to get a spot in a mathematics or philosophy department.

Speaker at a podium: 'If one person kills someone, 50% of the people involved are victims. If 99 people kill someone, 1% of the people involved are victims. The percent of victims is given by V = the limit of K/x as x approachs infinity, where K is people killed and x is the number of people killed. Thus, for sufficiently large x, murder is a victimless crime. So, the bigger we make a war, the more ethical it becomes!'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th of March, 2020. I have many essays that mention something raised by this comic strip. The many things Saturday Morning Breakfast Cereal has given me to write about are at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th is a less heady topic. Its speaker is doing an ethical calculation. These sorts of things are easy to spin into awful conclusions. They treat things like suffering with the same tools that we use to address the rates of fluids mixing, or of video game statistics. This often seems to trivialize suffering, which we feel like we shouldn’t do.

This kind of calculation is often done, though. It’s rather a hallmark of utilitarianism to try writing an equation for an ethical question. It blends often more into economics, where the questions can seem less cruel even if they are still about questions of life and death. But as with any model, what you build into the model directs your results. The lecturer here supposes that guilt is diminished by involving more people. (This seems rather true to human psychology, though it’s likely more that the sense of individual responsibility dissolves in a large enough group. There are many other things at work, though, all complicated and interacting in nonlinear ways.) If we supposed that the important measure was responsibility for the killing, we would get that the more people involved in killing, the worse it is, and that a larger war only gets less and less ethical. (This also seems true to human psychology.)

Mamet: 'I figure I have about 14,000 remaining days of life. So what's the big deal if I want to spend ONE of those days goofing off? That still leaves me with 13,00 days!' Cobb: 'Maybe you could spend a couple of those days learning math.' Mamet: 'Wait, make that 12,000. I'll need one day to PLAN the goof-off day.'
Jeff Corriveau’s Deflocked for the 20th of March, 2020. I’m surprised to learn this is a new tag for me. I’ve discussed the strip, it appears, only twice before, in 2012 and in 2015, before I tagged strips by name. All right. Well, this and future appearances by Deflocked will be at this link.

Jeff Corriveau’s Deflocked for the 20th sees Mamet calculating how many days of life he expects to have left. There are roughly 1,100 days in three years, so, Mamet’s figuring on about 40 years of life. These kinds of calculation are often grim to consider. But we all have long-term plans that we would like to do (retirement, and its needed savings, are an important one) and there’s no making a meaningful plan without an idea of what the goals are.


This finally closes out the last week’s comic strips. Please stop in next week as I get to some more mathematics comics and the Playful Math Education Blog Carnival. Thanks for reading.

Reading the Comics, March 17, 2020: Random Edition


I thought last week’s comic strips mentioning mathematics in detail were still subjects easy to describe in one or two paragraphs each. I wasn’t quite right. So here’s a half of a week, even if it is a day later than I had wanted to post.

John Zakour and Scott Roberts’s Working Daze for the 15th is a straggler Pi Day joke, built on the nerd couple Roy and Kathy letting the date slip their minds. This is a very slight Pi Day reference but I feel the need to include it for completeness’s sake. It reminds me of the sequence where one year Schroeder forgot Beethoven’s birthday, and was devastated.

Sue: 'So, Roy, what big fun did you and Kathy have for Pi Day this year?' Roy, caught by surprise, freezes, and then turns several colors in succession before he starts to cry. Ed, to Sue: 'Hard to say which is worse for him, that you forgot, or that you remembered.'
John Zakour and Scott Roberts’s Working Daze for the 15th of March, 2020. Essays featuring Working Daze, which often turns up in Pi Day events, are at this link. And generally essays tied to Pi Day are at this link.

Lincoln Peirce’s Big Nate for the 15th is a wordy bit of Nate refusing the story problem. Nate complains about a lack of motivation for the characters in it. But then what we need for a story problem isn’t the characters to do something so much as it is the student to want to solve the problem. That’s hard work. Everyone’s fascinated by some mathematical problems, but it’s hard to think of something that will compel everyone to wonder what the answer could be.

At one point Nate wonders what happens if Todd stops for gas. Here he’s just ignoring the premise of the question: Todd is given as travelling an average 55 mph until he reaches Saint Louis, and that’s that. So this question at least is answered. But he might need advice to see how it’s implied.

Quiz: 'Many lives in Los Angeles. Todd lives in Boston. They plan to meet in St Louis, which is 1,825 miles from Los Angeles and 1,192 miles from Boston. If Mandy takes a train travelling a constant 80 mph and Todd drives a car at a constant 55 mph, which of them will reach St Lous first?' Nate's answer: 'That depends. Who ARE these people? Are they a couple? Is this romance? If it is, wouldn't Todd drive way faster than 55 mph? He'd be all fired up to see Many, right? And wouldn't Mandy take a plane and get to St Louis in like three hours? Especially if she hasn't seen Todd in a while? But we don't know how long since they've been together because you decided not to tell us! Plus anything can happen while they're traveling. What if Todd stops for gas and the cashier is a total smoke show and he's like, Mandy Who? I can't answer until I have some real intel on these people. I can't believe you even asked the question.' Out loud, 'Also, Todd and Mandy are dorky names.' Teacher: 'This isn't what I meant by show your work.'
Lincoln Peirce’s Big Nate for the 15th of March, 2020. Essays with something mentioned by either Big Nate or the 1990s-repeats Big Nate: First Class are gathered at this link.

So this problem is doable by long division: 1825 divided by 80, and 1192 divided by 55, and see what’s larger. Can we avoid dividing by 55 if we’re doing it by hand? I think so. Here’s what I see: 1825 divided by 80 is equal to 1600 divided by 80 plus 225 divided by 80. That first is 20; that second is … eh. It’s a little less than 240 divided by 80, which is 3. So Mandy will need a little under 23 hours.

Is 23 hours enough for Todd to get to Saint Louis? Well, 23 times 55 will be 23 times 50 plus 23 times 5. 23 times 50 is 22 times 50 plus 1 times 50. 22 times 50 is 11 times 100, or 1100. So 23 times 50 is 1150. And 23 times 5 has to be 150. That’s more than 1192. So Todd gets there first. I might want to figure just how much less than 23 hours Mandy needs, to be sure of my calculation, but this is how I do it without putting 55 into an ugly number like 1192.

Cow: 'What're you doing?' Billy: 'I'm devising a system to win the lottery! Plugging in what I know about chaos theory and numerical behavior in nonlinear dynamical systems should give me the winning picks.' (Silent penultimate panel.) Cow: 'You're just writing down a bunch of numbers.' Billy: 'Maybe.'
Mark Leiknes’s Cow and Boy repeat for the 17th of March, 2020. The too-rare appearances of Cow and Boy Reruns in my essays are here.

Mark Leiknes’s Cow and Boy repeat for the 17th sees the Boy, Billy, trying to beat the lottery. He throws at it the terms chaos theory and nonlinear dynamical systems. They’re good and probably relevant systems. A “dynamical system” is what you’d guess from the name: a collection of things whose properties keep changing. They change because of other things in the collection. When “nonlinear” crops up in mathematics it means “oh but such a pain to deal with”. It has a more precise definition, but this is its meaning. More precisely: in a linear system, a change in the initial setup makes a proportional change in the outcome. If Todd drove to Saint Louis on a path two percent longer, he’d need two percent more time to get there. A nonlinear system doesn’t guarantee that; a two percent longer drive might take ten percent longer, or one-quarter the time, or some other weirdness. Nonlinear systems are really good for giving numbers that look random. There’ll be so many little factors that make non-negligible results that they can’t be predicted in any useful time. This is good for drawing number balls for a lottery.

Chaos theory turns up a lot in dynamical systems. Dynamical systems, even nonlinear ones, often have regions that behave in predictable patterns. We may not be able to say what tomorrow’s weather will be exactly, but we can say whether it’ll be hot or freezing. But dynamical systems can have regions where no prediction is possible. Not because they don’t follow predictable rules. But because any perturbation, however small, produces changes that overwhelm the forecast. This includes the difference between any possible real-world measurement and the real quantity.

Obvious question: how is there anything to study in chaos theory, then? Is it all just people looking at complicated systems and saying, yup, we’re done here? Usually the questions turn on problems such as how probable it is we’re in a chaotic region. Or what factors influence whether the system is chaotic, and how much of it is chaotic. Even if we can’t say what will happen, we can usually say something about when we can’t say what will happen, and why. Anyway if Billy does believe the lottery is chaotic, there’s not a lot he can be doing with predicting winning numbers from it. Cow’s skepticism is fair.

T-Rex: 'Dromiceiomimus, pick a number between one and a hundred thousand million.' Dromiceiomimus: '17?' T-Rex: 'Gasp! That's the number I was thinking of!' Dromiceiomimus: 'Great! Do I win something?' T-Rex: 'You just came out on a one in a hundred thousand million chance and you want a prize? It's not enough to spit in the face of probability itself?' Utahraptor: 'It's not THAT unlikely she'd chose your number. We're actually pretty bad at random number generation and if you ask folks to pick a number in a range, some choices show up more often than others. It's not that unlikely you'd both land on the same number!' T-Rex: 'But *I* didn't choose 17 randomly! It's ... the number of times I have thought about ice cream today, I'm not even gonna lie.'
Ryan North’s Dinosaur Comics for the 17th of March, 2020. Essays that mention something brought up in Dinosaur Comics are gathered at this link.

Ryan North’s Dinosaur Comics for the 17th is one about people asked to summon random numbers. Utahraptor is absolutely right. People are terrible at calling out random numbers. We’re more likely to summon odd numbers than we should be. We shy away from generating strings of numbers. We’d feel weird offering, say, 1234, though that’s as good a four-digit number as 1753. And to offer 2222 would feel really weird. Part of this is that there’s not really such a thing as “a” random number; it’s sequences of numbers that are random. We just pick a number from a random sequence. And we’re terrible at producing random sequences. Here’s one study, challenging people to produce digits from 1 through 9. Are their sequences predictable? If the numbers were uniformly distributed from 1 through 9, then any prediction of the next digit in a sequence should have a one chance in nine of being right. It turns out human-generated sequences form patterns that could be forecast, on average, 27% of the time. Individual cases could get forecast 45% of the time.

There are some neat side results from that study too, particularly that they were able to pretty reliably tell the difference between two individuals by their “random” sequences. We may be bad at thinking up random numbers but the details of how we’re bad can be unique.


And I’m not done yet. There’s some more comic strips from last week to discuss and I’ll have that post here soon. Thanks for reading.

Reading the Comics, March 20, 2020: Running from the Quiz Edition


I’m going to again start the week with the comics that casually mentioned mathematics. Later in the week I’ll have ones that open up discussion topics. I just don’t want you to miss a comic where a kid doesn’t want to do a story problem.

John Graziano’s Ripley’s Believe It or Not for the 15th mentions the Swiss mint issuing a tiny commemorative coin of Albert Einstein. I mention just because Einstein is such a good icon for mathematical physics.

Ashleigh Brilliant’s Pot-Shots for the 16th has some wordplay about multiplication and division. I’m not sure it has any real mathematical content besides arithmetic uniting multiplication and division, though.

Mark Pett’s Mr Lowe rerun for the 17th has the students bored during arithmetic class. Fractions; of course it would be fractions.

Justin Boyd’s Invisible Bread for the 18th> has an exhausted student making the calculation of they’ll do better enough after a good night’s sleep to accept a late penalty. This is always a difficult calculation to make, since you make it when your thinking is clouded by fatigue. But: there is no problem you have which sleep deprivation makes better. Put sleep first. Budget the rest of your day around that. Take it from one who knows and regrets a lot of nights cheated of rest. (This seems to be the first time I’ve mentioned Invisible Bread around here. Given the strip’s subject matter that’s a surprise, but only a small one.)

John Deering’s Strange Brew for the 18th is an anthropomorphic-objects strip, featuring talk about mathematics phobia.

One of Gary Larson’s The Far Side reruns for the 19th is set in a mathematics department, and features writing a nasty note “in mathematics”. There are many mathematical jokes, some of them written as equations. A mathematician will recognize them pretty well. None have the connotation of, oh, “Kick Me” or something else that would belong as a prank sign like that. Or at least nobody’s told me about them.

Tauhid Bondia’s Crabgrass for the 20th sees Kevin trying to find luck ahead of the mathematics quiz.

Bob Weber Jr and Jay Stephens’s Oh, Brother! for the 20th similarly sees Bud fearing a mathematics test.


Thanks for reading. And, also, please remember that I’m hosting the Playful Math Education Blog Carnival later this month. Please share with me any mathematics stuff you’ve run across that teaches or entertains or more.

Reading the Comics, March 14, 2020: Pi Day Edition


Pi Day was observed with fewer, and fewer on-point, comic strips than I had expected. It’s possible that the whimsy of the day has been exhausted. Or that Comic Strip Master Command advised people that the educational purposes of the day were going to be diffused because of the accident of the calendar. And a fair number of the strips that did run in the back half of last week weren’t substantial. So here’s what did run.

Gary Wise and Lance Aldrich’s Real Life Adventures for the 12th has a parent complaining about kids being allowed to use calculators to do mathematics. The rejoinder, asking how good they were at mathematics anyway, is a fair one.

Bill Watterson’s Calvin and Hobbes rerun for the 13th sees Calvin avoiding his mathematics homework. The strip originally ran the 16th of March, 1990.

And now we get to the strips that actually ran on the 14th of March.

Gracie, to her father: 'If I had $1.39 for every time I've struggled with a mathematics problem ... I'd have ... ' (She taps on a calculator) '6.23 cents.'
Hector D Cantú and Carlos Castellanos’s Baldo for the 14th of March, 2020. Essays with some mention of Baldo are gathered at this link.

Hector D Cantú and Carlos Castellanos’s Baldo is a slightly weird one. It’s about Gracie reflecting on how much she’s struggled with mathematics problems. There are a couple pieces meant to be funny here. One is the use of oddball numbers like 1.39 or 6.23 instead of easy-to-work-with numbers like “a dollar” or “a nickel” or such. The other is that the joke is .. something in the vein of “I thought I was wrong once, but I was mistaken”. Gracie’s calculation indicates she thinks she’s struggled with a math problem a little under 0.045 times. It’s a peculiar number. Either she’s boasting that she struggles very little with mathematics, or she’s got her calculations completely wrong and hasn’t recognized it. She’s consistently portrayed as an excellent student, though. So the “barely struggles” or maybe “only struggles a tiny bit at the start of a problem” interpretation is more likely what’s meant.

Mark Parisi’s Off the Mark is a Pi Day joke that actually features π. It’s also one of the anthropomorphic-numerals variety of jokes. I had also mistaken it for a rerun. Parisi’s used a similar premise in previous Pi Day strips, including one in 2017 with π at the laptop.

An anthropomorphic pi at a laptop, facing a web page demanding, 'Enter your full name'. It's gotten through 26 digits past the decimal.
Mark Parisi’s Off The Mark for the 14th of March, 2020. Other essays featuring something raised by Off The Mark, including a fair number of Pi Day jokes, are at this link.

π has infinitely many decimal digits, certainly. Of course, so does 2. It’s just that 2 has boring decimal digits. Rational numbers end up repeating some set of digits. It can be a long string of digits. But it’s finitely many, and compared to an infinitely long and unpredictable string, what’s that? π we know is a transcendental number. Its decimal digits go on in a sequence that never ends and never repeats itself fully, although finite sequences within it will repeat. It’s one of the handful of numbers we find interesting for reasons other than their being transcendental. This though nearly every real number is transcendental. I think any mathematician would bet that it is a normal number, but we don’t know that it is. I’m not aware of any numbers we know to be normal and that we care about for any reason other than their normality. And this, weirdly, also despite that we know nearly every real number is normal.

At the ATM, a pie with arms enters a pin. An onlooking doughnut says '3.14? Please tell me that's not really your pin.'
Dave Whamond’s Reality Check for the 14th of March, 2020. Essays that show off something from a Reality Check panel are at this link.

Dave Whamond’s Reality Check plays on the pun between π and pie, and uses the couple of decimal digits of π that most people know as part of the joke. It’s not an anthropomorphic numerals joke, but it is circling that territory.

Loose sketch of Albert Einstein, accompanied by the quote, 'Only two things are infinite: the universe and human stupidity, and I'm not sure about the former', along with a note wishing him a happy birthday.
Michael Cavna’s Warped for the 14th of March, 2020. The rare appearances here of Warped are gathered at this link.

Michael Cavna’s Warped celebrates Albert Einstein’s birthday. This is of marginal mathematics content, but Einstein did write compose one of the few equations that an average lay person could be expected to recognize. It happens that he was born the 14th of March and that’s, in recent years, gotten merged into Pi Day observances.


I hope to start discussing this week’s comic strips in some essays starting next week, likely Sunday. Thanks for reading.

Reading the Comics, March 11, 2020: Half Week Edition


There were a good number of comic strips mentioning mathematical subjects last week, as you might expect for one including the 14th of March. Most of them were casual mentions, though, so that’s why this essay looks like this. And is why the week will take two pieces to finish.

Jonathan Lemon and Joey Alison Sayer’s Little Oop for the 8th is part of a little storyline for the Sunday strips. In this the young Alley Oop has … travelled in time to the present. But different from how he does in the weekday strips. What’s relevant about this is Alley Oop hearing the year “2020” and mentioning how “we just got math where I come from” but being confident that’s either 40 or 400. Which itself follows up a little thread in the Sunday strips about new numbers on display and imagining numbers greater than three.

Venn Diagram with two bubbles. The left is 'Day after Daylight Savings [sic] Start'; the right is 'Monday'. The intersection has an arrow from it pointing to a travel cup of coffee.
Maria Scrivan’s Half Full for the 9th of March, 2020. Essays featuring some topic raised by Half Full appear at this link.

Maria Scrivan’s Half Full for the 9th is the Venn Diagram strip for the week.

Paul Trap’s Thatababy for the 9th is a memorial strip to Katherine Johnson. She was, as described, a NASA mathematician, and one of the great number of African-American women whose work computing was rescued from obscurity by the book and movie Hidden Figures. NASA, and its associated agencies, do a lot of mathematical work. Much of it is numerical mathematics: a great many orbital questions, for example, can not be answered with, like, the sort of formula that describes how far away a projectile launched on a parabolic curve will land. Creating a numerical version of a problem requires insight and thought about how to represent what we would like to know. And calculating that requires further insight, so that the calculation can be done accurately and speedily. (I think about sometime doing a bit about the sorts of numerical computing featured in the movie, but I would hardly be the first.)

Eulogy strip, as drawn by the baby, celebrating Katherine Johnson, NASA mathematician 1918 - 2020. It shows a child's drawing of her, and of a Mercury capsule, with formulas describing a ballistic trajectory making the motion trail of the capsule.
Paul Trap’s Thatababy for the 9th of March, 2020. My essays featuring something raised by Thatababy are at this link.

I also had thought the Mathematical Moments from the American Mathematical Society had posted an interview with her last year. I was mistaken but in, I think, a forgivable way. In the episode “Winning the Race”, posted the 12th of June, they interviewed Christine Darden, another of the people in the book, though not (really) the movie. Darden joined NASA in the late 60s. But the interview does talk about this sort of work, and how it evolved with technology. And, of course, mentions Johnson and her influence.

Graham Harrop’s Ten Cats for the 9th is another strip mentioning Albert Einstein and E = mc2. And using the blackboard full of symbols to represent deep thought.

Patrick Roberts’s Todd the Dinosaur for the 10th showcases Todd being terrified of fractions. And more terrified of story problems. I can’t call it a false representation of the kinds of mathematics that terrify people.

Teacher: 'All right, class, please take out your math books!' Todd: 'Teacher, this isn't gonna be fractions, is it?' Teacher: 'No, Todd, no fractions.' Todd: 'Whewwww!' Teacher: 'Now listen carefully, class. Train A leaves Chicago at 7:00 am, and ... ' (Todd, screaming in panic, runs out crashing through the wall and over the horizon.)
Patrick Roberts’s Todd the Dinosaur for the 10th of March, 2020. Essays that discuss something mentioned in a Todd the Dinosaur should be gathered at this link.

Stephen Beals’s Adult Children for the 11th has a character mourning that he took calculus as he’s “too stupid to be smart”. Knowing mathematics is often used as proof of intelligence. And calculus is used as the ultimate of mathematics. It’s a fair question why calculus and not some other field of mathematics, like differential equations or category theory or topology. Probably it’s a combination of slightly lucky choices (for calculus). Calculus is old enough to be respectable. It’s often taught as the ultimate mathematics course that people in high school or college (and who aren’t going into a mathematics field) will face. It’s a strange subject. Learning it requires a greater shift in thinking about how to solve problems than even learning algebra does. And the name is friendly enough, without the wordiness or technical-sounding language of, for example, differential equations. The subject may be well-situated.

Tony Rubino and Gary Markstein’s Daddy’s Home for the 11th has the pacing of a logic problem, something like the Liar’s Paradox. It’s also about homework which happens to be geometry, possibly because the cartoonists aren’t confident that kids that age might be taking a logic course.


I’ll have the rest of the week’s strips, including what Comic Strip Master Command ordered done for Pi Day, soon. And again I mention that I’m hosting this month’s Playful Math Education Blog Carnival. If you have come across a web site with some bit of mathematics that brought you delight and insight, please let me know, and mention any creative projects that you have, that I may mention that too. Thank you.

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.

Quincy, playing with a basketball: 'Li'l Bo, some pro basketball players have million-dollar contracts.' Bo; 'Boy! That's a lot of money for playing a game.' Quincy: 'I figure they're gettin' about a dollar a dribble.'
Ted Shearer’s Quincy for the 5th of March, 2020. It looks to have originally run the 9th of January, 1981. I don’t get to discuss the strip often, but when I do, Quincy appears in essays at this link.

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, February 29, 2020: Leap Day Quiet Edition


I can clear out all last week’s mathematically-themed comic strips in one move, it looks like. There were a fair number of strips; it’s just they mostly mention mathematics in passing.

Bill Amend’s FoxTrot for the 23rd — a new strip; it’s still in original production for Sundays — has Jason asking his older sister to double-check a mathematics problem. Double-checking work is reliably useful, as proof against mistakes both stupid and subtle. But that’s true of any field.

Mark Tatulli’s Heart of the City for the 23rd has Heart preparing for an algebra test.

Jim Unger’s Herman for the 23rd has a parent complaining about the weird New Math. The strip is a rerun and I don’t know from when; it hardly matters. The New Math has been a whipping boy for mathematics education since about ten minutes after its creation. And the complaint attaches to every bit of mathematics education reform ever. I am sympathetic to parents, who don’t see why their children should be the test subjects for a new pedagogy. And who don’t want to re-learn mathematics in order to understand what their children are doing. But, still, let someone know you were a mathematics major and they will tell you how much they didn’t understand or like mathematics in school. It’s hard to see why not try teaching it differently.

(If you do go out pretending to be a mathematics major, don’t worry. If someone challenges you on a thing, cite “Euler’s Theorem”, and you’ll have said something on point. And I’ll cover for you.)

Phil Dunlap’s Ink Pen rerun for the 24th has Bixby Rat complain about his mathematics skills.

Father and child duck sitting on the starry sky. Father: 'Hey, Champ, I know you're only 5, but I think it's time I introduce you to the wonders of the universe! See those stars? How many do you think there are?' Child: 'Um ... 12?' Father: 'Actually, there's over 300 sextillion stars! That's a 3 with 23 zeroes after it.' Child: 'And that's more than 12?' Father: 'Maybe I should introduce you to the wonders of math, first.'
Brian Gordon’s Fowl Language for the 25th of February, 2020. This strip previously ran the 5th of February, 2016, which happens to be the only other time I have an essay mentioning this comic. That’s from before I tagged comic strips by title, though. So this essay and any future repetitions that happen to mention Fowl Language should be at this link, although the previous one probably won’t be.

Brian Gordon’s Fowl Language for the 25th has a father trying to explain the vastness of Big Numbers to their kid. Past a certain point none of us really know how big a thing is. We can talk about 300 sextillion stars, or anything else, and reason can tell us things about that number. But do we understand it? Like, can we visualize that many stars the way we can imagine twelve stars? This gets us into the philosophy of mathematics pretty soundly. 300 sextillion is no more imaginary than four is, but I know I feel more confident in my understanding of four. How does that make sense? And can you explain that to your kid?

Vic Lee’s Pardon my Planet for the 28th has an appearance by Albert Einstein. And a blackboard full of symbols. The symbols I can make out are more chemistry than mathematics, but they do exist just to serve as decoration.

Bud Blake’s Tiger rerun for the 28th has Hugo mourning his performance on a mathematics test.

Ruben Bolling’s Super-Fun-Pak Comix for the 28th is an installment of The Uncertainty Principal. This is a repeat, even allowing that Super-Fun-Pak Comix are extracted reruns from Tom The Dancing Bug. As I mention in the essay linked there, the uncertainty principle being referred to here is a famous quantum mechanics result. It tells us there are sets of quantities whose values we can’t, even in principle, measure simultaneously to unlimited precision. A precise measurement of, for example, momentum destroys our ability to be precise about position. This is what makes the joke here. The mathematics of this reflects non-commutative sets of operators.

Dave Blazek’s Loose Parts for the 29th is another with a blackboard full of symbols used to express deep thought on a subject.


And that takes care of last week. I’ll be Reading the Comics for their mathematics content next week, too, although the start of the week has been a slow affair so far. We’ll see if that changes any.

Reading the Comics, February 21, 2020: February 21, 2020 Edition


So way back about fifty years ago, when pop science started to seriously explain how computers worked, and when the New Math fad underscored how much mathematics is an arbitrary cultural choice, the existence of number bases other than ten got some publicity. This offered the chance for a couple of jokes, or at least things which read to pop-science-fans as jokes. For example, playing on a typographical coincidence between how some numbers are represented in octal (base eight) and decimal (base ten), we could put forth this: for computer programmers Halloween is basically another Christmas. After all, 31 OCT = 25 DEC. It’s not much of a joke, but how much of a joke could you possibly make from “writing numbers in different bases”? Anyway, Isaac Asimov was able to make a short mystery out of it.

Tony Cochrane’s Agnes for the 21st is part of a sequence with Agnes having found some manner of tablet computer. Automatic calculation has always been a problem in teaching arithmetic. A computer’s always able to do more calculations, more accurately, than a person is; so, whey do people need to learn anything about how to calculate? The excuse that we might not always have a calculator was at least a little tenable up to about fifteen years ago. Now it’d take a massive breakdown in society for computing devices not to be pretty well available. This would probably take long enough for us to brush up on long division.

Teacher: 'Agnes, take out your math book.' Agnes: 'No need. I now own a semi-educational, quasi-computer electronic pad or something. If I boop enough buttons in the correct sequence, all world info will be there to behold! Including all the indecipherable doggerel *you're* pushing.' [ At the Principal's Office ] Agnes: 'Math teachers are fans of big numbers ... not so much big words.'
Tony Cochrane’s Agnes for the 21st of February, 2020. Essays exploring something mentioned in Agnes appear at this link.

It’s more defensible to say that people need to be able to say whether an answer is plausible. If we don’t have any expectations for the answer, we don’t know whether we’ve gone off and calculated a wrong thing. This is a bit more convincing. We should have some idea whether 25, 2500, or 25 million is the more likely answer. That won’t help us spot whether we made a mistake and got 27 instead of 25, though. It does seem reasonable to say that we can’t appreciate mathematics, so much of which is studying patterns and structures, without practicing. And arithmetic offers great patterns and structures, while still being about things that we find familiar and useful. So that’s likely to stay around.

Miss Prunelly wincing. Jughaid has written on the board '2 + 7 = baseball team', '5 + 6 = football team', and '4 + 1 = basketball team'. Jughaid says 'Gosh, Miz Prunelly, these are easy!' The other students laugh.
John Rose’s Barney Google and Snuffy Smith for the 21st of February, 2020. The occasional strip which mentions Barney Google and Snuffy Smith appears at this link. Google’s in the strip now for one or two weeks a year.

John Rose’s Barney Google and Snuffy Smith for the 21st is a student-subverting-the-blackboard-problem joke. Jughaid’s put the arithmetic problems into terms of what he finds most interesting. To me, it seems like if this is helping him get comfortable with the calculations, let him. If he does this kind of problem often enough, he’ll get good at it and let the false work of going through sports problems fade away.

Pig, reading 'Retirement Calculator: To determine your annual retirement income, just do the following: add your total personal savings to your total employee pension. Divide by the number of retirement years you plan to enjoy.' He works out: 0 + 0 / 0 = 0. Pig, to Goat: 'I love when the math is easy.'
Stephan Pastis’s Pearls Before Swine for the 21st of February, 2020. Essays featuring some point raised by Pearls Before Swine are gathered at this link. No, I don’t know why his every Sunday strip is complaining about the perilously perilous peril of political correctness anymore. I agree it feels like he’s trying to get ahead of something, but, like, he’s got a buffer of like seven years ahead of publication. If he’s got something he’s going to be expected to apologize for you’d think we’d have heard rumors or something by now.

Stephan Pastis’s Pearls Before Swine for the 21st sees Pig working through a simple Retirement Calculator. He appreciates the mathematics being easy. A realistic model would have wrinkles to it. For example, the retirement savings would presumably be returning interest, from investments or from simple deposit accounts. Working out how much one gets from that, combined with possibly spending down the principal, can be involved. But a rough model doesn’t need this sort of detailed complication. It can be pretty simple, and still give you some guidance to what a real answer should look like.

Caption: 'You may be a GEEK if ... you think that doing math in hexadecimal will impress the ladies.' Jay, at a bar, saying, 'Yeah, it's interesting when ya think about it, but 1A + 2B = 45 ... '; two women, walking away, roll their eyes and think of a dripping faucet.
John Zakour and Scott Roberts’s Working Daze for the 21st of February, 2020. This strip doesn’t get a lot of attention from me outside of Pi Day, but when it does, Working Daze gets a mention at this link.

John Zakour and Scott Roberts’s Working Daze for the 21st is a joke about how guys assuming that stuff they like is inherently interesting to other people. In this case, it’s hexadecimal arithmetic. That’s at least got the slight appeal that we’ve settled on using a couple of letters as numerals for it, so that wordplay and word-like play is easier than it is in base ten.


And this wraps up a string of comic strips all with some mathematical theme that all posted on the same day. I grant none of these get very deep into mathematical topics; that’s all right. There’ll be some more next week in a post at this link. Thank you.

Reading the Comics, February 19, 2020: 90s Doonesbury Edition


The weekday Doonesbury has been in reruns for a very long while. Recently it’s been reprinting strips from the 1990s and something that I remember producing Very Worried Editorials, back in the day.

Garry Trudeau’s Doonesbury for the 17th reprints a sequence that starts off with the dread menace and peril of Grade Inflation, the phenomenon in which it turns out students of the generational cohort after yours are allowed to get A’s. (And, to a lesser extent, the phenomenon in which instructors respond to the treatment of education as a market by giving the “customers” the grades they’re “buying”.) The strip does depict an attitude common towards mathematics, though, the idea that it must be a subject immune to Grade Inflation: “aren’t there absolute answers”? If we are careful to say what we mean by an “absolute answer” then, sure.

Dean: 'Sir, you're going to have to speak to the faculty about grade inflation. Standards are just falling off the chart. The pressure to pander is even beginning to affect the math department.' President: 'Math? How can that be? Aren't there absolute answers in math?' Dean: 'Well, yes and no.' President, thinking: 'Yes and now?' [ Math Class ] Student: '17!' Other Student: '39!' Math Professor: 'Excellent guesses! Well done!'
Garry Trudeau’s Doonesbury rerun for the 17th of February, 2020 of February, 2020. It originally ran the 20th of December, 1993. I have few essays which mention this long-running strip, oddly. What essays are inspired by something in Doonesbury appear at this link.

But grades? Oh, there is so much subjectivity as to what goes into a course. And into what level to teach that course at. How to grade, and how harshly to grade. It may be easier, compared to other subjects, to make mathematics grading more consistent year-to-year. One can make many problems that test the same skill and yet use different numbers, at least until you get into topics like abstract algebra where numbers stop being interesting. But the factors that would allow any course’s grade to inflate are hardly stopped by the department name.

Mathematician: 'I went massively into debt to build a machine that generates holographic numbers and equations whenever I wish to appear thoughtful.' Friend: 'Was that a good use money?' [ Panel of the mathematician looking thoughtful with equations spread out in space behind and in front of her. ] Mathematician: 'Yes.' Friend: 'A thousand times yes.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th of February, 2020. I have a few essays which don’t mention this long-running web strip, oddly. What essays are inspired by something in Saturday Morning Breakfast Cereal appear at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th is a strip about using a great wall of equations as emblem of deep, substantial thought. The equations depicted are several meaningful ones. The top row is from general relativity, the Einstein Field Equations. These relate the world-famous Ricci curvature tensor with several other tensors, describing how mass affects the shape of space. The P = NP line describes a problem of computational science with an unknown answer. It’s about whether two different categories of problems are, in fact, equivalent. The line about L = -\frac{1}{4} F_{\mu \nu} F^{\mu \nu} is a tensor-based scheme to describe the electromagnetic field. The next two lines look, to me, like they’re deep in Schrödinger’s Equation, describing quantum mechanics. It’s possible Weinersmith has a specific problem in mind; I haven’t spotted it.

Guy Walks Into A Bar comics. Man holding a horse's reins, to the bartender: 'I'll bet $50 my horse can do arithmetic!' Bartender: 'OK, what's 2 + 2?' Horse: '10.' Horse, to the angry guy, outside the bar: 'Well, think about it. Why would a horse use base ten?'
Ruben Bolling’s Super-Fun-Pak Comix for the 18th of February, 2020. There are a fair number of essays inspired by one of the Super-Fun-Pak Comics, and they’re gathered at this link. All the Super-Fun-Pak Comics first ran in Tom The Dancing Bug, essays about which appear here.

Ruben Bolling’s Super-Fun-Pak Comix for the 18th is one of the Guy Walks Into A Bar line, each of which has a traditional joke setup undermined by a technical point. In this case, it’s the horse counting in base four, in which representation the number 2 + 2 is written as 10. Really, yes, “10 in base four” is the number four. I imagine properly the horse should say “four” aloud. But it is quite hard to read the symbols “10” as anything but ten. It’s not as though anyone looks at the hexadecimal number “4C” and pronounces it “76”, either.

Garry Trudeau’s Doonesbury for the 19th twisted the Grade Inflation peril to something that felt new in the 90s: an attack on mathematics as “Eurocentric”. The joke depends on the reputation of mathematics as finding objectively true things. Many mathematicians accept this idea. After all, once we’ve seen a proof that we can do the quadrature of a lune, it’s true regardless of what anyone thinks of quadratures and lunes, and whether that person is of a European culture or another one.

Student: 'This B+ is wrong, man! You're dissin me big time here.' Professor: 'Mr Slocum, I merely gave you the grade you deserved.' Student: 'Can't be, man! This is WAY off base!' Professor: 'As was your entire first proof, in which you held the square root of 144 to be 15. It is, in fact, 12.' Student: 'Well, sure, from a narrow, absolutist, Eurocentric perspective, maybe it's 12.' Professor: 'So?' Student: 'So my culture teaches it's 15, man!' Professor: 'Fascinating. Would this be an advanced civilization?'
Garry Trudeau’s Doonesbury rerun for the 19th of February, 2020 of February, 2020. It originally ran the 22nd of December, 1993. I am reminded once again of a fellow grad student, doing his teaching-assistant duties, watching student after student on the calculus exam reduce 1002 to 10. When enough students make the same mistake you start to question your grading scheme. Which is sometimes fair: if everyone gets partway through a question and fails at the same step there’s a prima facie case that the problem was your instruction, not their comprehension. Doesn’t cover dumb arithmetic glitches, though.

But there are several points to object to here. The first is, what’s a quadrature? … This is a geometric thing; it’s finding a square that’s the same area as some given shape, using only straightedge and compass constructions. The second is, what’s a lune? It’s a crescent moon-type shape (hence the name) that you can make by removing the overlap from two circles of specific different radiuses arranged in a specific way. It turns out you can find the quadrature for the lune shape, which makes it seem obvious that you should be able to find the quadrature for a half-circle, a way easier (to us) shape. And it turns out you can’t. The third question is, who cares about making squares using straightedge and compass? And the answer is, well, it’s considered a particularly elegant way of constructing shapes. To the Ancient Greeks. And to those of us who’ve grown in a mathematics culture that owes so much to the Ancient Greeks. Other cultures, ones placing more value on rulers and protractors, might not give a fig about quadratures and lunes.

This before we get into deeper questions. For example, if we grant that some mathematical thing is objectively true, independent of the culture which finds it, then what role does the proof play? It can’t make the thing more or less true. It doesn’t eve matter whether the proof is flawed, or whether it convinces anyone. It seems to imply a mathematician isn’t actually needed for their mathematics. This runs contrary to intuition.

Anyway, this gets off the point of the student here, who’s making a bad-faith appeal to multiculturalism to excuse laziness. It’s difficult to imagine a culture that doesn’t count, at least, even if they don’t do much work with numbers like 144. Granted that, it seems likely they would recognize that 12 has some special relationship with 144, even if they don’t think too much of square roots as a thing.


And do please stop in later this Leap Day week. I figure to have one of my favorite little things, a Reading the Comics day that’s all one day. It should be at this link, when posted. Thank you.

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.

Gunther: 'OK, let's see what you did wrong here on number 26.' Luann notices Aaron Hill walking past, and goes out to follow him. Meanwhile Gunther works out 'Simplify: 6x + 7(3 + x + 4)'. After Luann's gone through several rooms following Aaron, Gunther calls out, 'The answer is 13x + 49!' Luann: 'What? Oh! OK, thanks!'
Greg Evans’s Luann Againn for the 12th of February, 2020. The strip originally ran the 12th of February, 1992. Essays mentioning something inspired by Luann, either the current run or the 1992-vintage Luann Againn reruns, are at this link.

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.

Woman, looking at people buying lottery tickets: 'I feel sorry for them.' Cow: 'Why?' Woman: 'Because, statistically, their chances are so slim that they're wasting their money.' Later, Cow: 'Let's go play!' Woman: 'Can't, need to work.' Cow: 'Why?' Woman: 'Because I want to become a successful artist and give my family a good life.' Cow: 'You know, statistically, your chance is so slim that you are wasting your time.' This word balloon stabs the woman between the eyes.
Chen Weng’s Messycow Comics for the 13th of February, 2020. The occasional essay mentioning something raised by Messycow Comics appear at this link.

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.

Teacher: 'So what is 3 times 55?' Looking out over a bunch of students, many with hands up. One with her hand way up, several feet taller than anyone else's. Gracie's hand is this; she's got a fake extra-long arm on a stick and waves that. Other students near her look at her and glare.
Hector D. Cantü and Carlos Castellanos’s Baldo for the 14th of February, 2020. Essays featuring something mentioned by Baldo appear at this link.

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, February 11, 2020: Symbols Edition


Finally we get to last week’s comics. This past one wasn’t nearly so busy a week for mathematically-themed comic strips. But there’s still just enough that I can split them across two days. This fits my schedule well, too.

Rick Detorie’s One Big Happy for the 9th is trying to be the anthropomorphized numerals joke of the week. It’s not quite there, but it also uses some wordplay. … And I’ll admit being impressed any of the kids could do much with turning any of the numerals into funny pictures. I remember once having a similar assignment, except that we were supposed to use the shape of our state, New Jersey, as the basis for the picture. I grant I am a dreary and literal-minded person. But there’s not much that the shape of New Jersey resembles besides itself, “the shape of Middlesex County, New Jersey”, and maybe a discarded sock. I’m not still upset about this.

Parents night at the school. On the wall are assignments: 'Make a funny picture drawing using a numeral', with kids who've drawn 2 as a dog or 0 as a clown or 8 as a snowman or such. Ruthie's drawn 5 as a figure with a cap and a bindle walking away. Ruthie's Mom 'I liked your drawing, Ruthie, the 'Five' running away from home.' Ruthie: 'Oh yeah, my roamin' numeral!'
Rick Detorie’s One Big Happy for the 9th of February, 2020. Essays exploring something from One Big Happy, current (creators.com) or rerun (gocomics.com) runs, are at this link.

Samson’s Dark Side of the Horsefor the 11th is another on the counting-sheep theme. It’s built on the resemblance between the numeral ‘2’ and the choice of ‘z’ to represent sleeping.

Horace, counting sheep: '222,220' as a sheep staggers past the imagined fence. '222,221' as a sheep barely climbs over the fence. '222,222' as the sheep, and Horace, collapses flat into sleep.
Samson’s Dark Side of the Horsefor the 11th of February, 2020. This and other essays featuring Dark Side of the Horse trying to sleep are at this link.

The choice of ‘z’ to mean a snore is an arbitrary choice, no more inherent to the symbol than that ‘2’ should mean two. Christopher Miller’s American Cornball, which tracks a lot of (American) comedic conventions of the 20th century, notes a 1911 comic postcard representing snoring as “Z-Z-Z-Z-R-R-R-R-Z-Z-Z-Z-R-R-R-R”, which captures how the snore is more than a single prolonged sound.

Dave Blazek’s Loose Parts for the 11th has the traditional blackboard full of symbols. And two mathematics-types agreeing that they could make up some more symbols. Well, mathematics is full of symbols. Each was created by someone. Each had a point, which was to express some concept better. Usually the goal is to be more economical: it’s fewer strokes of the pen to write = instead of “equals”, and = is quicker even than “eq”. Or we want to talk a lot about a complicated concept, which is how we get, say, \sin^{-1} x for “a representative of the set of angles with sine equal to x”.

Two figures in front of a board full of symbols. One says: 'I think you're right, John. If we can come up with one new nonsense symbol a week, we can stretch this gig out for, like, a year.'
Dave Blazek’s Loose Parts for the 11th of February, 2020. Essays featuring some discussion of Loose Parts are at this link.

I suspect every mathematician has made up a couple symbols in their notes. In the excitement of working out a problem there’ll be something they want to refer to a lot. That gets reduced to an acronym or a repeated scribble soon enough. Sometimes it’s done by accident: for a while when I needed a dummy variable I would call on “ksee”, a Greek letter so obscure that it does not even exist. It looks like a cross between zeta and xi. The catch is, always, getting anyone else to use the symbol. Most of these private symbols stay private, because they don’t do work that can’t be better done by a string of symbols we already have (letters included). Or at least they don’t to well enough to be worth the typesetting trouble. I’d be surprised if any of the students I used “ksee” in front of reused the letter, even if they did find a need for a dummy variable. Founding a field, or writing a definitive text in a field, helps your chances.

I am curious how the modern era of digital typesetting will affect symbol creation. It’s relatively easy to put in a new symbol — or to summon one in the Unicode universe not currently used for mathematics — in a document and have it copied. Certainly it’s easy compared to what it was like in typewriter and Linotype days, when you might need to rely on a friend who knows a guy at the type foundry. On the other hand, it’s hard enough to get the raw file in LaTeX — a long-established standard mathematics typesetting computer language — from another person and have it actually work, even without adding in new symbols. I don’t see that changing just because several people have found that a bubble tea emoji quite helps their paper on sedimentation rates.

A long multipanel story called 'Girls Win', about the contest of boys versus girls. The relevant section starts with the narration, 'Even at school, I knew boys always lose to girls.' Teacher: 'OK, children, we're going to play Math Baseball' (a game played on the chalkboard by getting three problems right.) 'Girls were just smarter somehow.' Teacher: 'Let's separate into boys vs girls.' 'Even though we had math-wiz Sergio on our team, he alone couldn't save us from ourselves.' Teacher: 'Strike three! Yer out!' Teammate, berating Pedro, who's missed 11 - 9: 'Dang it, Pedro! We had this!'
Pedro Martin’s Mexikid Stories for the 11th of February, 2020. I haven’t had cause to discuss this strip before, so it’s a new tag. But this and any future essays mentioning Mexikid Stories should be at this link.

Pedro Martin’s Mexikid Stories for the 11th recounts childhood memories and anxieties of being matched, boys versus girls, in various activities. This includes mathematics quizzes. Here, the mathematics is done as a class game, which is a neat coincidence as I’d been thinking of similar public mathematics quiz-games that I’d done. I liked them, but then, I was almost always at top or second in the class rankings, and — after the initial couple rounds — never fell below third. My recent thoughts were for how much less fun this must have been for the kids in 26th place, especially if they’re ones who can do the work just fine, given time and space. We do value speed, in working, and that comes from practicing a task so often that we do it in the slightest time possible. And we value ability to perform under pressure, so we put people into anxiety-producing states until they can do a particular task anyway.


Thanks for reading. I should have another post at this link, most likely Thursday.

Reading the Comics, February 8, 2020: Delta Edition


With this essay, I finally finish the comic strips from the first full week of February. You know how these things happen. I’ll get to the comics from last week soon enough, at an essay gathered under this link. For now, some pictures with words:

Art Sansom and Chip Sansom’s The Born Loser for the 7th builds on one of the probability questions people often use. That is the probability of an event, in the weather forecast. Predictions for what the weather will do are so common that it takes work to realize there’s something difficult about the concept. The weather is a very complicated fluid-dynamics problem. It’s almost certainly chaotic. A chaotic system is deterministic, but unpredictable, because to get a meaningful prediction requires precision that’s impossible to ever have in the real world. The slight difference between the number π and the number 3.1415926535897932 throws calculations off too quickly. Nevertheless, it implies that the “chance” of snow on the weekend means about the same thing as the “chance” that Valentinte’s Day was on the weekend this year. The way the system is set up implies it will be one or the other. This is a probability distribution, yes, but it’s a weird one.

Gladys: 'I wonder what the weather will be like this weekend.' Brutus; 'The TV forecaster says there's less than a 10% chance of snow! Of course, that forecaster has less than a 10% chance of being correct!'
Art Sansom and Chip Sansom’s The Born Loser for the 7th of February, 2020. When I discuss something raised by The Born Loser I put the essay at this link.

What we talk about when we say the “chance” of snow or Valentine’s on a weekend day is one of ignorance. It’s about our estimate that the true value of something is one of the properties we find interesting. Here, past knowledge can guide us. If we know that the past hundred times the weather was like this on Friday, snow came on the weekend less than ten times, we have evidence that suggests these conditions don’t often lead to snow. This is backed up, these days, by numerical simulations which are not perfect models of the weather. But they are ones that represent something very like the weather, and that stay reasonably good for several days or a week or so.

And we have the question of whether the forecast is right. Observing this fact is used as the joke here. Still, there must be some measure of confidence in a forecast. Around here, the weather forecast is for a cold but not abnormally cold week ahead. This seems likely. A forecast that it was to jump into the 80s and stay there for the rest of February would be so implausible that we’d ignore it altogether. A forecast that it would be ten degrees (Fahrenheit) below normal, or above, though? We could accept that pretty easily.

Proving a forecast is wrong takes work, though. Mostly it takes evidence. If we look at a hundred times the forecast was for a 10% chance of snow, and it actually snowed 11% of the time, is it implausible that the forecast was right? Not really, not any more than a coin coming up tails 52 times out of 100 would be suspicious. If it actually snowed 20% of the time? That might suggest that the forecast was wrong. If it snowed 80% of the time? That suggests something’s very wrong with the forecasting methods. It’s hard to say one forecast is wrong, but we can have a sense of what forecasters are more often right than others are.

Caption; 'When I hear two dogs barking ... ' And the picture shows one dog going 'Arf! Arf! Arf', interrupted by a dog barking. Then the first dog goes, 'Woof! Woof! Arf! Arf' Caption: ' ... I like to imagine that one of them is trying to count, while the other is yelling out random numbers.' First dog: '36 ... 37 ... 38' Second Dog: '72!' First Dog: 'Dude! Stop it! 1 ... 2 ... '
Doug Savage’s Savage Chickens for the 7th of February, 2020. Essays that mention something based on Savage Chickens are put at this link.

Doug Savage’s Savage Chickens for the 7th is a cute little bit about counting. Counting things out is an interesting process; for some people, hearing numbers said aloud will disrupt their progress. For others, it won’t, but seeing numbers may disrupt it instead.

Scientist types, standing in a room full of dogs, with a right triangle diagram on the wall. Scientist: 'Now we have proof, Wickingham! If you show this image to 500 golden retrievers every day for ten years, they are UNABLE to discover Pythagoras's Theorem.'
Niklas Eriksson’s Carpe Diem for the 8th of February, 2020. The occasional essay based on something mentioned in Carpe Diem is gathered at this link.

Niklas Eriksson’s Carpe Diem for the 8th is a bit of silliness about the mathematical sense of animals. Studying how animals understand number is a real science, and it turns up interesting results. It shouldn’t be surprising that animals can do a fair bit of counting and some geometric reasoning, although it’s rougher than even our untrained childhood expertise. We get a good bit of our basic mathematical ability from somewhere, because we’re evolved to notice some things. It’s silly to suppose that dogs would be able to state the Pythagorean Theorem, at least in a form that we recognize. But it is probably someone’s good research problem to work out whether we can test whether dogs understand the implications of the theorem, and whether it helps them go about dog work any.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th speaks of the “Cinnamon Roll Delta Function”. The point is clear enough on its own. So let me spoil a good enough bit of fluff by explaining that it’s a reference to something. There is, lurking in mathematical physics, a concept called the “Dirac delta function”, named for that innovative and imaginative fellow Paul Dirac. It has some weird properties. Its domain is … well, it has many domains. The real numbers. The set of ordered pairs of real numbers, R2. The set of ordered triples of real numbers, R3. Basically any space you like, there’s a Dirac delta function for it. The Dirac delta function is equal to zero everywhere in this domain, except at one point, the “origin”. At that one function, though? There it’s equal to …

Graph: 'The Cinnamon Roll Delta Function.' y-axis: tastiness. x-axis: quality of ingredients. For a long stretch of quality the taste is at zero: 'tastes like dry bread with sugar.' Then the vertical spike. After that, the taste is zero again: 'Why is there fennel and orange blossom? Did I strange my inner child?'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th of February, 2020. If you don’t see an essay mentioning this strip, wait five minutes. Or look at my collection of Saturday Morning Breakfast Cereal-inspired discussions, here.

Here we step back a moment. We really, really, really want to say that it’s infinitely large at that point, which is what Weinersmith’s graph shows. If we’re being careful, we don’t say that though. Because if we did say that, then we would lose the thing that we use the Dirac delta function for. The Dirac delta function, represented with δ, is a function with the property that for any set D, in the domain, that you choose to integrate over

\int_D \delta(x) dx = 1

whenever the origin is inside the interval of integration D. It’s equal to 0 if the origin is not inside the interval of integration. This, whatever the set is. If we use the ordinary definitions for what it means to integrate a function, and say that the delta function is “infinitely big” at the origin, then this won’t happen; the integral will be zero everywhere.

This is one of those cases where physicists worked out new mathematical concepts, and the mathematicians had to come up with a rationalization by which this made sense. This because the function is quite useful. It allows us, mathematically, to turn descriptions of point particles into descriptions of continuous fields. And vice-versa: we can turn continuous fields into point particles. It turns out we like to do this a lot. So if we’re being careful we don’t say just what the Dirac delta function “is” at the origin, only some properties about what it does. And if we’re being further careful we’ll speak of it as a “distribution” rather than a function.

But colloquially, we think of the Dirac delta function as one that’s zero everywhere, except for the one point where it’s somehow “a really big infinity” and we try to not look directly at it.

The sharp-eyed observer may notice that Weinersmith’s graph does not put the great delta spike at the origin, that is, where the x-axis represents zero. This is true. We can create a delta-like function with a singular spot anywhere we like by the process called “translation”. That is, if we would like the function to be zero everywhere except at the point a , then we define a function \delta_a(x) = \delta(x - a) and are done. Translation is a simple step, but it turns out to be useful all the time.

Thanks again for reading. See you soon.

Reading the Comics, February 3, 2020: Fake Venn Diagrams and Real Reruns Edition


Besides kids doing homework there were a good ten or so comic strips with enough mathematical content for me to discuss. So let me split that over a couple of days; I don’t have the time to do them all in one big essay.

Sandra Bell-Lundy’s Between Friends for the 2nd is declared to be a Venn Diagram joke. As longtime readers of these columns know, it’s actually an Euler Diagram: a Venn Diagram requires some area of overlap between all combinations of the various sets. Two circles that never touch, or as these two do touch at a point, don’t count. They do qualify as Euler Diagrams, which have looser construction requirements. But everything’s named for Euler, so that’s a less clear identifier.

Caption: 'The Venn Diagram of the Sandwich Generation.' Two tangent circles, one 'The Problem' and one 'The Solution'. Two friends sit pondering this over coffee 'It's what put the 'vent' in 'venti'.'
Sandra Bell-Lundy’s Between Friends for the 2nd of February, 2020. Essays mentioning Between Friends and its imperfectly formed Venn Diagrams are at this link.

John Kovaleski’s Daddy Daze for the 2nd talks about probability. Particularly about the probability of guessing someone’s birthday. This is going to be about one chance in 365, or 366 in leap years. Birthdays are not perfectly uniformly distributed through the year. The 13th is less likely than other days in the month for someone to be born; this surely reflects a reluctance to induce birth on an unlucky day. Births are marginally more likely in September than in other months of the year; this surely reflects something having people in a merry-making mood in December. These are tiny effects, though, and to guess any day has about one chance in 365 of being someone’s birthday will be close enough.

Toddler, pointing: 'Ba ba ba.' Dad: 'Her? ... Excuse me, my son would like to give you something.' Woman: 'Uh ... OK?' Dad: 'It's a birthday card he made.' Woman: 'But it's not my birthday. ... It's ... lovely.' Dad: 'He likes to give them out to random people. He figures the odds are 1 in 365 it'll be someone's birthday and it'll make them happy.' Woman: 'What are the odds it won't be someone's birthday and it'll still make them happy?'
John Kovaleski’s Daddy Daze for the 2nd of February, 2020. Essays which mention something from Daddy Daze should be at this link.

If the child does this long enough there’s almost sure to be a match of person and birthday. It’s not guaranteed in the first 365 cards given out, or even the first 730, or more. But, if the birthdays of passers-by are independent — one pedestrian’s birthday has nothing to do with the next’s — then, overall, about one-365th of all cards will go to someone whose birthday it is. (This also supposes that we won’t see things like the person picked saying that while it’s not their birthday, it is their friend’s, here.) This, the Law of Large Numbers, one of the cornerstones of probability, guarantees us.

Conference room. Projected on a wall is 'Diplopia', represented by two overlapping circles. Man at the table asks: 'Is everyone seeing a Venn diagram, or just me?'
Mark Anderson’s Andertoons for the 2nd of February, 2020. Some of the many essays mentioning Andertoons are at this link.

Mark Anderson’s Andertoons for the 2nd is the Mark Anderson’s Andertoons for the week. And it’s a Venn Diagram joke, at least if the two circles are “really” there. Diplopia is what most of us would call double vision, seeing multiple offset copies of a thing. So the Venn diagram might be an optical illusion on the part of the businessman and the reader.

Man in the Accounting department, to a person entering: 'Hey, c'mon in, Warren. We were just crunching a few numbers.' Another person is jumping up and down on a 2, a 3, and has broken several other numerals.
Brian Boychuk and Ron Boychuk’s Chuckle Brothers repeat for the 3rd of February, 2020. It originally ran the 22nd of February, 2011. Essays featuring some aspect of The Chuckle Brothers are at this link.

Brian Boychuk and Ron Boychuk’s Chuckle Brothers for the 3rd is not quite the anthropomorphic numerals joke of the week. At least, it’s built on manifesting numerals and doing things with them.

Letters 'x' and 'y' sit at a bar. The y says, 'I just knew that someday, our paths would intersect.'
Dave Blazek’s Loose Parts for the 3rd of February, 2020. Essays with some mention of topics raised by Loose Parts are at this link.

Dave Blazek’s Loose Parts for the 3rd is an anthropomorphic mathematical symbols joke. I suppose it’s algebraic symbols. We usually get to see the ‘x’ and ‘y’ axes in (high school) algebra, used to differentiate two orthogonal axes. The axes can be named anything. If ‘x’ and ‘y’ won’t do, we might move to using \hat{i} and \hat{j} . In linear algebra, when we might want to think about Euclidean spaces with possibly enormously many dimensions, we may change the names to \hat{e}_1 and \hat{e}_2 . (We could use subscripts of 0 and 1, although I do not remember ever seeing someone do that.)

Mikki: 'First our teacher says 6 and 4 make 10. Then she says 7 and 3 equals 10. Then 5 and 5 make 10. We need a teacher who can make up her mind.'
Morrie Turner’s Wee Pals repeat for the 3rd of February, 2020. Many, but not all, of the essays featuring Wee Pals are at this link.

Morrie Turner’s Wee Pals for the 3rd is a repeat, of course. Turner died several years ago and no one continued the strip. But it is also a repeat that I have discussed in these essays before, which likely makes this a good reason to drop Wee Pals from my regular reading here. There are 42 distinct ways to add (positive) whole numbers up to make ten, when you remember that you can add three or four or even six numbers together to do it. The study of how many different ways to make the same sum is a problem of partitioning. This might not seem very interesting, but if you try to guess how many ways there are to add up to 9 or 11 or 15, you’ll notice it’s a harder problem than it appears.


And for all that, there’s still some more comic strips to review. I will probably slot those in to Sunday, and start taking care of this current week’s comic strips on … probably Tuesday. Please check in at this link Sunday, and Tuesday, and we’ll see what I do.

Reading the Comics, February 8, 2020: Exams Edition


There were a bunch of comic strips mentioning some kind of mathematical theme last week. I need to clear some out. So I’ll start with some of the marginal mentions. Many of these involve having to deal with exams or quizzes.

Jonathan Mahood’s Bleeker: The Rechargeable Dog from the 3rd started a sequence about the robot dog helping Skip with his homework. This would include flash cards, which weren’t helping, in preparation for a test. Bleeker would go to slightly ridiculous ends, since, after all, you never know when something will click.

Bleeker extends his arms, cupping them together in a square shape. Skip: 'Do you think this will help me figure out the square root of these numbers?' Bleeker: 'We should try everything, Skip.'
Jonathan Mahood’s Bleeker: The Rechargeable Dog for the 8th of February, 2020. Essays that mention something brought up by Bleeker appear at this link.

There are different ways to find square roots. (I can guarantee that Skip wasn’t expected to use this one.) The term ‘root’ derives from an idea that the root of a number is the thing that generates it: 3 is a square root of 9 because multiplying 3’s together gives you 9. ‘Square’ is I have always only assumed because multiplying a number by itself will give you the area of a square with sides of length that number. This is such an obvious word origin, though, that I am reflexively suspicious. Word histories are usually subtle and capricious things.

Bill Watterson’s Calvin and Hobbes for the 3rd began the reprint of a storyling based on a story-problem quiz. Calvin fantasizes solving it in a wonderful spoof of hardboiled detective stories. There is a moment of Tracer Bullet going over exactly what information he has, which is a good first step for any mathematics problem. I assume it’s also helpful for solving real mysteries.

Calvin, narrating as Tracer Bullet, wandering through inky, rain-soaked city streets at night: 'I stepped out into the rainy streets and reviewed the facts. There weren't many. Two saps, Jack and Joe, drive toward each other at 60 and 30 mph. After 10 minutes, they pass. I'm supposed to find out how far apart they started. Questions pour down like the rain. Who ARE these mugs? What are they trying to accomplish? Why was Jack in such a hurry? And what difference does it make where they started from? I had a hunch that, before this was over, I'd be sorry I asked.'
Bill Watterson’s Calvin and Hobbes rerun for the 5th of February, 2020. It originally ran the 7th of February, 1990. Essays inspired by something in Calvin and Hobbes should be at this link. I appreciate this all the more since I got into old-time radio, and could imagine the narration in the cadence of specific shows. This is more remarkable since Watterson’s claimed he didn’t care about the hardboiled detective genre and was just spoofing stuff he had picked up elsewhere. It speaks to Watterson’s writing skills that this spoof-based-on-spoofs still feels funny. Of course, he’s helped by how if anything is off, that’s all right, since it’s in the voice of a seven-year-old.

The strip for the 8th closing the storyline has a nice example of using “billion” as a number so big as to be magical, capable of anything. Big numbers can do strange and contrary-to-intuition things. But they can be reasoned out.

Tony Cochran’s Agnes for the 4th sees the title character figuring she could sell her “personal smartness”. Her best friend Trout wonders if that’s tutoring math or something. (Incidentally, Agnes is one of the small handful of strips to capture what made Calvin and Hobbes great; I recommend giving it a try.)

Bill Amend’s FoxTrot Classics reprint for the 6th mentions that Peter has a mathematics test scheduled, and shows part of his preparation.

Charlie Brown, looking at the problems 7 + 6 = and 4 - 3 = on the board: 'Why do I always get the hardest problems? Let's see. If our team had 7, and we scored a touchdown but failed to convert, we'd have 13. And if par on a hole is four, and you get a birdie, you're one under.' He walks away, having successfully done 7 + 6 = 13 and 4 - 3 = 1.
Charles Schulz’s Peanuts Begins for the 5th of February, 2020. The strip originally ran the 6th of February, 1952. The other strip ran the 9th of February that year. And appearances by Peanuts or Peanuts Begins should be in essays at this link. (Peanuts Begins reprints comics from the 1950s. The ordinary run of Peanuts is reprinting strips, this year, from 1973.)

Charles Schulz’s Peanuts Begins for the 5th sees Charlie Brown working problems on the board. He’s stuck for what to do until he recasts the problem as scoring in football and golf. We may giggle at this, but I support his method. It’s convinced him the questions are worth solving, the most important thing to doing them at all. And it’s gotten him to the correct answers. Casting these questions as sports problems is the building of falsework: it helps one do the task, and then is taken away (or hidden) from the final product. Everyone who does mathematics builds some falsework like this. If we do a particular problem, or kind of problem, often enough we get comfortable enough with the main work that we don’t need the falsework anymore. So it is likely to be for Charlie Brown.

On the 8th is another strip of Charlie Brown doing arithmetic in class. Here he just makes a mistake from having counted in a funny way all morning. This, too, happens to us all.


I will have more Reading the Comics posts at this link, hopefully this week. Incidentally other essays mentioning Agnes are at this link, and essays mentioning FoxTrot, reruns or the new-run Sundays, are here. Thanks for reading.

Reading the Comics, February 1, 2020: I Never Talk About Marvin Edition


There’s some comic strips that get mentioned here all the time. Then there’s comic strips that I have been reading basically my whole life, and that never give me a thread to talk about. Although I’ve been reading comic strips for their mathematics content for a long while now, somehow, I am still surprised when these kinds of comic strip are not the same thing. So here’s the end of last week’s comics, almost in time for next week to start:

Kevin Fagan’s Drabble for the 28th has Penny doing “math” on colors. Traditionally I use an opening like this to mention group theory. In that we study things that can be added together, in ways like addition works on the integers. Colors won’t quite work like this, unfortunately. A group needs an element that’s an additive identity. This works like zero: it can be added to anything without changing its value. There isn’t a color that you can mix with other colors that leaves the other color unchanged, though. Even white or clear will dilute the original color.

Mom: 'How was school today, Penny?' Penny: 'Great, Mommy! I learned how to do math! Want me to show you? Blue plus red equals purple!'
Kevin Fagan’s Drabble for the 28th of January, 2020. It doesn’t come up often, but when it does, Drabble appears in essays at this link.

If you’ve thought of the clever workaround, that each color can be the additive identity to itself, you get credit for ingenuity. Unfortunately, to be a group there has to be a lone additive identity. Having more than one makes a structure that’s so unlike the integers that mathematicians won’t stand for it. I also don’t know of any interesting structures that have more than one additive identity. This suggests that nobody has found a problem that they represent well. But the strip suggests maybe it could tell us something useful for colors. I don’t know.

Marvin: 'After all the talk about 'fake news' I'm starting to question EVERYTHING big people tell me.' He's looking at a teacher holding up the flashcard 1 + 1 = 2.
Tom Armstrong’s Marvin for the 28th of January, 2020. I don’t think it has ever come up before, but what the heck. Any essays which mention Marvin should be at this link.

Tom Armstrong’s Marvin for the 28th is a strip which follows from the discovery that “fake news” is a thing that people say. Here the strip uses a bit of arithmetic as the sort of incontrovertibly true thing that Marvin is dumb to question. Well, that 1 + 1 equals 2 is uncontrovertibly true, unless we are looking at some funny definitions of ‘1’ or ‘plus’ or something. I remember, as a kid, being quite angry with a book that mentioned “one cup of popcorn plus one cup of water does not give us two cups of soggy popcorn”, although I didn’t know how to argue the point.

Title: 'The Math Homework.' Dad, in the kitchen, to kid: 'What's surface area? Ask your mother.' The mother is in the kitchen, working, and has every bit of surface area that isn't being used for homework with cooking tools. Footer joke: Mom asks, 'Can you please move? I need this space.'
Hilary Price and Rina Piccolo’s Rhymes with Orange for the 30th of January, 2020. Essays with some mention of Rhymes With Orange should be at this link.

Hilary Price and Rina Piccolo’s Rhymes with Orange for the 30th is … well, I’m in this picture and I don’t like it. I come from a long line of people who cover every surface with stuff. But as for what surface area is? … Well, there’s a couple of possible definitions. One that I feel is compelling is to think of covering sets. Take a shape that’s set, by definition, to have an area of 1 unit of area. What is the smallest number of those unit shapes which will cover the original shape? Cover is a technical term here. But also, here, the ordinary English word describes what we need it for. How many copies of the unit shape do you need to exactly cover up the whole original shape? That’s your area. And this fits to the mother’s use of surfaces in the comic strip neatly enough.

Mutt: 'What's the matter, you stuck?' Jeff, looking at his car: 'Yes and no! I tried the cary products they advertise on TV. They claimed this car would use 50% less gas. Then I bought a carburettor which saves 30%, special spark plugs which save 20% and a new brand of gas which saved 10%! Now when I drive the gas tank overflows!' Jeff shows gas pouring out of the tank.
Bud Fisher’s Mutt and Jeff for the 31st of January, 2020. And the essays which have mentioned Mutt and Jeff comics appear at this link.

Bud Fisher’s Mutt and Jeff for the 31st is a rerun of vintage unknown to me. I’m not sure whether it’s among the digitally relettered strips. The lettering’s suspiciously neat, but, for example, there’s at least three different G’s in there. Anyway, it’s an old joke about adding together enough gas-saving contraptions that it uses less than zero gas. So far as it’s tenable at all, it comes from treating percentage savings from different schemes as additive, instead of multiplying together. Also, I suppose, that the savings are independent, that (in this case) Jeff’s new gas saving ten percent still applies even with the special spark plugs or the new carburettor [sic]. The premise is also probably good for a word problem, testing out understanding of percentages and multiplication, which is just a side observation here.


This wraps up last week’s mathematically-themed comic strips. This week I can tell you already was a bonanza week. When I start getting to its comics I should have an essay at this link. Thanks for reading.

Reading the Comics, January 27, 2020: Alley Oop Followup Edition


I apologize for missing Sunday. I wasn’t able to make the time to write about last week’s mathematically-themed comic strips. But I’m back in the swing of things. Here are some of the comic strips that got my attention.

Jonathan Lemon and Joey Alison Sayers’s Little Oop for the 26th has something neat in the background. Oop and Garg walk past a vendor showing off New Numbers. This is, among other things, a cute callback to one of the first of Lemon and Sayers’s Little Oop strips.. (And has nothing to do with the daily storyline featuring the adult Alley Oop.) And it is a funny idea to think of “new numbers”. I imagine most of us trust that numbers are just … existing, somewhere, as concepts independent of our knowing them. We may not be too sure about the Platonic Forms. But, like, “eight” seems like something that could plausibly exist independently of our understanding of it.

Science Expo. Little Alley Oop leads Garg past the New Numbers stand to the Multistick. Garg: 'A stick? That sounds boring.' Vendor, holding up a stick: 'Quite the opposite, young man! The multi-stick can do everything! You can use it as a weapon, you can light it on fire and use it as a torch, you can use it as a fishing pole. It has literally dozens of uses!' Garg: 'Can I use it as a toy for my pet dinosaur?' Vendor: 'Well, I wouldn't recommend it. We haven't tested it out for that.' Garg: 'Eh, no thanks.'
Jonathan Lemon and Joey Alison Sayers’s Little Oop for the 26th of January, 2020. The handful of times I’ve head to talk about Alley Oop or Little Oop are gathered at this link.

Still, we do keep discovering things we didn’t know were numbers before. The earliest number notations, in the western tradition, for example, used letters to represent numbers. This did well for counting numbers, up to a large enough total. But it required idiosyncratic treatment if you wanted to handle large numbers. Hindu-Arabic numerals make it easy to represent whole numbers as large as you like. But that’s at the cost of adding ten (well, I guess eight) symbols that have nothing to do with the concept represented. Not that, like, ‘J’ looks like the letter J either. (There is a folk etymology that the Arabic numerals correspond to the number of angles made if you write them out in a particular way. Or less implausibly, the number of strokes needed for the symbol. This is ingenious and maybe possibly has helped one person somewhere, ever, learn the symbols. But it requires writing, like, ‘7’ in a way nobody has ever done, and it’s ahistorical nonsense. See section 96, on page 64 of the book and 84 of the web presentation, in Florian Cajori’s History of Mathematical Notations.)

Still, in time we discovered, for example, that there were irrational numbers and those were useful to have. Negative numbers, and those are useful to have. That there are complex-valued numbers, and those are useful to have. That there are quaternions, and … I guess we can use them. And that we can set up systems that resemble arithmetic, and work a bit like numbers. Those are often quite useful. I expect Lemon and Sayers were having fun with the idea of new numbers. They are a thing that, effectively, happens.

Francis, answering the phone: 'Hi, Nate Yeah, I did the homework. No, I'm not giving you the answers. ... I'm sure you did try hard ... I know it's due tomorrow ... You're not going to learn anything if I just ... of course I don't want to get in trouble but ... all right! This once! For #1, I got 4.5. For #2, I got 13.3. For #3, I got ... hello?' Cut to Nate, hanging up the phone: 'Wrong number.' Nate's Dad: 'I'll say.'
Lincoln Peirce’s Big Nate: First Class for the 26th of January, 2020. It originally ran the 15th of January, 1995. Essays mentioning either Big Nate or the rerun Big Nate: First Class should be gathered at this link.

Lincoln Peirce’s Big Nate: First Class for the 26th has Nate badgering Francis for mathematics homework answers. Could be any subject, but arithmetic will let Peirce fit in a couple answers in one panel.

Other Man: 'Do you ever play the lottery?' Brutus: 'I believe your chances of winning the lottery are the same as your chances of being struck by lightning!' Other: 'Have I told you the time I bought an instant lottery ticket on a whim? I won one thousand dollars!' Brutus: 'No kidding? That changes everything I said about the odds! That must've been the luckiest day of your life!' Other: 'Not really; as I left the store, I was struck by lightning!'
Art Sansom and Chip Sansom’s The Born Loser for the 26th of January, 2020. There are times that I discuss The Born Loser, and those essays are at this link.

Art Sansom and Chip Sansom’s The Born Loser for the 26th is another strip on the theme of people winning the lottery and being hit by lightning. And, as I’ve mentioned, there is at least one person known to have won a lottery and survived a lightning strike.

Woman: 'How's the project coming?' Boy: 'Fine.' Quiet panel. Then, a big explosion. Woman: 'I thought you guys were doing math!' Girl: 'Engineering!' Boy: 'It's *like* math, but louder.'
David Malki’s Wondermark for the 27th of January, 2020. I am surprised to learn that I already have a tag for this comic, but it turns out I’ve mentioned it as long ago as late December. So, essays mentioning Wondermark: they’re at this link.

David Malki’s Wondermark for the 27th describes engineering as “like math, but louder”, which is a pretty good line. And it uses backgrounds of long calculations to make the point of deep thought going on. I don’t recognize just what calculations are being done there, but they do look naggingly familiar. And, you know, that’s still a pretty lucky day.

Wavehead at the chalkboard, multiplying 2.95 by 3.2 and getting, ultimately, to '.9.4.4.0.' He says: 'I forgot where to put the decimal, so I figured I'd cover all the bases.'
Mark Anderson’s Andertoons for the 27th of January, 2020. And I have a lot of essays mentioning something from Andertoons gathered at this link.

Mark Anderson’s Andertoons for the 27th is the Mark Anderson’s Andertoons for the week. It depicts Wavehead having trouble figuring where to put the decimal point in the multiplication of two decimal numbers. Relatable issue. There are rules you can follow for where to put the decimal in this sort of operation. But the convention of dropping terminal zeroes after the decimal point can make that hazardous. It’s something that needs practice, or better: though. In this case, what catches my eye is that 2.95 times 3.2 has to be some number close to 3 times 3. So 9.440 is the plausible answer.

Baseball dugout. One player: 'Jim makes $2.1 million per year. Fred makes $9.3 million over a three-year period. How much more does Fred make than Jim each year?' Second player: '60% of Roger's income last year came from promotional work. If his annual earnings are $17.2 million, how much of his income came just from baseball?' Third player: 'Tom was traded for two relief pitchers. If together they'll earn 1.3 times Tom's former annual yearly salary of $2.5 million, how much will each earn?'
Mike Twohy’s That’s Life for the 27th of January, 2020. So I have some essays mentioning this comic strip, but from before I started tagging them. I’ll try to add tags to those old essays when I have the chance. In the meanwhile, this essay and maybe future ones mentioning That’s Life should be at this link.

Mike Twohy’s That’s Life for the 27th presents a couple of plausible enough word problems, framed as Sports Math. It’s funny because of the idea that the workers who create events worth billions of dollars a year should be paid correspondingly.


This isn’t all for the week from me. I hope to have another Reading the Comics installment at this link, soon. Thanks for reading.

Reading the Comics, January 25, 2020: Comic Strip Master Command Is Making This Hard For Me Edition


Or they’re making it easy for me. But for another week all the comic strips mentioning mathematics have done so in casual ways. Ones that I don’t feel I can write a substantial paragraph about. And so, ones that I don’t feel I can fairly use the images of here. Here’s strips that at least said “math” somewhere in them:

Mark Pett’s Mr Lowe rerun for the 18th had the hapless teacher giving out a quiz about fractions.

Greg Cravens’s The Buckets for the 19th plays on the conflation of “zero” and “nothing”. The concepts are related, and we wouldn’t have a zero if we weren’t trying to worth with the concept of nothing. But there is a difference that’s quite hard to talk about without confusing matters.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 19th has a student accused of cheating on a pre-algebra test.

Liniers’s Macanudo for the 21st has a kid struggling with mathematics while the imaginary friend goes off and plays.

Lincoln Peirce’s Big Nate: First Class for the 21st has Nate struggling with mathematics. The strip is a reprint of the Big Nate from the 23rd of January, 1995.

Greg Curfman’s Meg for the 21st has Meg doing arithmetic homework.

Scott Hilburn’s The Argyle Sweater for the 23rd is a wordplay joke, with a flash card that has an addition problem on it.

One of Gary Larson’s The Far Side reprints for the 24th has a man demanding the answer to one question: the square root of an arbitrary number. It’s a little over 70, and that’s as far as anyone could reasonably expect to answer off the top of their head.

James Beutel’s Banana Triangle for the 24th quotes The Wizard Of Oz’s famous garbled version of the Pythagorean Theorem.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 25th presents a sinister reading of the fad of “prove you’re human” puzzles that demanded arithmetic expressions be done. All computer programs, including, like, Facebook group messages are arithmetic operations ultimately. The steps could be translated into simple expressions like this and be done by humans. It just takes work which, I suppose, could also be translated into other expressions.


And with that large pile of mentions I finish off the mathematical comic strips for the day. Also for the month: next Sunday gets us already into February. Sometime then I should post at this link a fresh Reading the Comics essay. Thank you for reading this one.

Reading the Comics, January 18, 2020: Decimals In Fractions Edition


Let me first share the other comic strips from last week which mentioned mathematics, but in a casual way.

Jerry Scott and Jim Borgman’s Zits for the 14th used the phrase “do the math”, and snarked on the younger generation doing mathematics. This was as part of the longrunning comic’s attempt to retcon the parents from being Baby Boomers to being Generation X. Scott and Borgman can do as they like but, I mean, their kids are named Chad and Jeremy. That’s only tenable if they’re Boomers. (I’m not sure Chad has returned from college in the past ten years.) And even then it was marginal.

John Kovaleski’s Bo Nanas rerun for the 14th is a joke about the probability of birthdays.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th features “the Bertrand Russell Drinking Game”, playing on the famous paradox about self-referential statements of logic.

Stephan Pastis’s Pearls Before Swine for the 17th has Rat use a bunch of mathematical jargon to give his declarations authority.

Cy Olson’s Office Hours for the 18th, rerunning a strip from the 9th of November, 1971, is in the line of jokes about parents not understanding their children’s arithmetic. It doesn’t seem to depend on mocking the New Math, which is a slight surprise for a 1971 comic.


Classroom. The blackboard problem is 0.25 / 0.05 = ? Wavehead, to teacher: 'Decimals *in* fractions?! Have you no shame?!'
Mark Anderson’s Andertoons for the 12th of January, 2020. This and other essays with some topic raised by Andertoons should appear at this link.

So Mark Anderson’s Andertoons for the 12th is the only comic strip of some substance that I noticed last week. You see what a slender month it’s been. It does showcase the unsettling nature of seeing notations for similar things mixed. It’s not that there’s anything which doesn’t parse about having decimals in the numerator or denominator. It just looks weird. And that can be enough to throw someone out of a problem. They might mistake the problem for one that doesn’t have a coherent meaning. Or they might mistake it for one too complicated to do. Learning to not be afraid of a problem that looks complicated is worth doing. As is learning how to tell whether a problem parses at all, even if it looks weird.


And that’s an end to last week in comics. I plan to have a fresh Reading the Comics post on Sunday. Thank you for reading in the meanwhile.