## Reading the Comics, August 10, 2019: In Security Edition

There were several more comic strips last week worth my attention. One of them, though, offered a lot for me to write about, packed into one panel featuring what comic strip fans call the Wall O’ Text.

Bea R’s In Security for the 9th is part of a storyline about defeating an evil “home assistant”. The choice of weapon is Michaela’s barrage of questions, too fast and too varied to answer. There are some mathematical questions tossed in the mix. The obvious one is “zero divided by two equals zero, but why’z two divided by zero called crazy town?” Like with most “why” mathematics questions there are a range of answers.

The obvious one, I suppose, is to appeal to intuition. Think of dividing one number by another by representing the numbers with things. Start with a pile of the first number of things. Try putting them into the second number of bins. How many times can you do this? And then you can pretty well see that you can fill two bins with zero things zero times. But you can fill zero bins with two things — well, what is filling zero bins supposed to mean? And that warns us that dividing by zero is at least suspicious.

That’s probably enough to convince a three-year-old, and probably most sensible people. If we start getting open-mined about what it means to fill no containers, we might say, well, why not have two things fill the zero containers zero times over, or once over, or whatever convenient answer would work? And here we can appeal to mathematical logic. Start with some ideas that seem straightforward. Like, that division is the inverse of multiplication. That addition and multiplication work like you’d guess from the way integers work. That distribution works. Then you can quickly enough show that if you allow division by zero, this implies that every number equals every other number. Since it would be inconvenient for, say, “six” to also equal “minus 113,847,506 and three-quarters” we say division by zero is the problem.

This is compelling until you ask what’s so great about addition and multiplication as we know them. And here’s a potentially fruitful line of attack. Coming up with alternate ideas for what it means to add or to multiply are fine. We can do this easily with modular arithmetic, that thing where we say, like, 5 + 1 equals 0 all over again, and 5 + 2 is 1 and 5 + 3 is 2. This can create a ring, and it can offer us wild ideas like “3 times 2 equals 0”. This doesn’t get us to where dividing by zero means anything. But it hints that maybe there’s some exotic frontier of mathematics in which dividing by zero is good, or useful. I don’t know of one. But I know very little about topics like non-standard analysis (where mathematicians hypothesize non-negative numbers that are not zero, but are also smaller than any positive number) or structures like surreal numbers. There may be something lurking behind a Quanta Magazine essay I haven’t read even though they tweet about it four times a week. (My twitter account is, for some reason, not loading this week.)

Michaela’s questions include a couple other mathematically-connected topics. “If infinity is forever, isn’t that crazy, too?” Crazy is a loaded word and probably best avoided. But there are infinity large sets of things. There are processes that take infinitely many steps to complete. Please be kind to me in my declaration “are”. I spent five hundred words on “two divided by zero”. I can’t get into that it means for a mathematical thing to “exist”. I don’t know. In any event. Infinities are hard and we rely on them. They defy our intuition. Mathematicians over the 19th and 20th centuries worked out fairly good tools for handling these. They rely on several strategies. Most of these amount to: we can prove that the difference between “infinitely many steps” and “very many steps” can be made smaller than any error tolerance we like. And we can say what “very many steps” implies for a thing. Therefore we can say that “infinitely many steps” gives us some specific result. A similar process holds for “infinitely many things” instead of “infinitely many steps”. This does not involve actually dealing with infinity, not directly. It involves dealing with large numbers, which work like small numbers but longer. This has worked quite well. There’s surely some field of mathematics about to break down that happy condition.

And there’s one more mathematical bit. Why is a ball round? This comes around to definitions. Suppose a ball is all the points within a particular radius of a center. What shape that is depends on what you mean by “distance”. The common definition of distance, the “Euclidean norm”, we get from our physical intuition. It implies this shape should be round. But there are other measures of distance, useful for other roles. They can imply “balls” that we’d say were octahedrons, or cubes, or rounded versions of these shapes. We can pick our distance to fit what we want to do, and shapes follow.

I suspect but do not know that it works the other way, that if we want a “ball” to be round, it implies we’re using a distance that’s the Euclidean measure. I defer to people better at normed spaces than I am.

Mark Anderson’s Andertoons for the 10th is the Mark Anderson’s Andertoons for the week. It’s also a refreshing break from talking so much about In Security. Wavehead is doing the traditional kid-protesting-the-chalkboard-problem. This time with an electronic chalkboard, an innovation that I’ve heard about but never used myself.

Bob Scott’s Bear With Me for the 10th is the Pi Day joke for the week.

And that last one seemed substantial enough to highlight. There were even slighter strips. Among them: Mark Anderson’s Andertoons for the 4th features latitude and longitude, the parts of spherical geometry most of us understand. At least feel we understand. Jim Toomey’s Sherman’s Lagoon for the 8th mentions mathematics as the homework parents most dread helping with. Larry Wright’s Motley rerun for the 10th does a joke about a kid being bad at geography and at mathematics.

And that’s this past week’s mathematics comics. Reading the Comics essays should all be gathered at this link. Thanks for reading this far.

## Reading the Comics, August 3, 2019: Summer Trip Edition

I was away from home most of last week. Comic Strip Master Command was kind and acknowledged this. There wasn’t much for me to discuss. There’s not even many comics too slight to discuss. I thank them for their work in not overloading me. But if you wondered why Sunday’s post was what it was, you now know. I suspect you didn’t wonder.

Mark Anderson’s Andertoons for the 29th of July is a comfortable and familiar face for these Reading the Comics posts. I’m glad to see it. The joke is built on negative numbers, and Wavehead’s right to say this is kind of the reason people hate mathematics. At least, that mathematicians will become comfortable with something that has a clear real-world intuitive meaning, such as that adding things together gets you a bigger thing. And then for good reasons of logic get to counter-intuitive things, such as adding things together to get a lesser thing. Negative numbers might be the first of these intuition-breaking things that people encounter. That or fractions. I encounter stories of people who refuse to accept that, say, $\frac16$ is smaller than $\frac13$, although I’ve never seen it myself.

So why do mathematicians take stuff like “adding” and break it? Convenience, I suppose, is the important reason. Having negative numbers lets us treat “having a quantity” and “lacking a quantity” using the same mechanisms. So that’s nice to have. If we have positive and negative numbers, then we can treat “adding” and “subtracting” using the same mechanisms. That’s nice to do. The trouble is then knowing, like, “if -3 times 4 is greater than -16, is -3 times -4 greater than 16? Or less than? Why?”

Jeffrey Caulfield and Brian Ponshock’s Yaffle for the 31st of July uses the blackboard-full-of-mathematics as shorthand for deep thought about topics. The equations don’t mean much of anything, individually or collectively. I’m curious whether Caulfield and Ponshock mean, in the middle there, for that equation to be π times y2 equalling z3, or whether it’s π times x times y2 that is. Doens’t matter either way. It’s just decoration.

And then there are the most marginal comic strips for the week. And if that first Yaffle didn’t count as too marginal to mention, think what that means for the others. Yaffle on the 28th of July features a mention of sudoku as the sort of thing one struggles to solve. Tony Rubino and Gary Markstein’s Daddy’s Home for the 1st of August mentions mathematics as the sort of homework a parent can’t help with. Jim Toomey’s Sherman’s Lagoon for the 2nd sets up a math contest. It’s mentioned as the sort of things the comic strip’s regular cast can’t hope to do.

And there we go. I’m ready now for August. Around Sunday I should have a fresh Reading the Comics page here. And it does seem like I’m missing my other traditional post here, doesn’t it? Have to work on that.

## Reading the Comics, October 6, 2018: Square Root of 144 Edition

And I have three last strips from last week to talk about. For those curious, I have ten comics for this week that I flagged for mention, at least before reading the Saturday GoComics pages. So that will probably be two or three installments next week. It’ll depend how many Saturday GoComics strips raise a point I feel like discussing.

Jim Toomey’s Sherman’s Lagoon for the 5th uses arithmetic as the archetypical homework problem that’s short enough to fit in a panel but also too hard for an adult to do. And, neatly, easy for a computer to do. Were I either shark here I’d have reasoned out the square root of 144 something like this: they’re not getting homework where they’d be asked the square root of something that wasn’t a perfect square. So it’s got to be a whole number. 144 is between 100 and 400, so it’s got to be the square root of something between 10 and 20. 144 is pretty close to 100, so 144’s square root is probably close to 10. The square of 1 is 1, so 11 squared has to be one-hundred-something-and-one. The square of 2 is 4, so 12 squared has to be one-hundred-something-and-four. The square of 3 is 9, so 13 squared has to be one-hundred-something-and-nine. The square of 4 is 16, so 14 squared has to be at least one-hundred-something-and-six. And by then we’re getting pretty far from 10. So the only plausible candidate is 12. Test that out and, what do you know, there it is.

Greg Cravens’s The Buckets for the 6th is a riff on the monkeys-at-keyboards joke. Well, what keeps monkeys-at-typewriters from writing interesting things is that they don’t have any selection. They just produce text to no end, in principle. Picking out characters and words that carry narrative is what makes essayists and playwrights. … That said, I think every instructor has faced the essay that is, somehow, worse than gibberish. The process is to try to find anything that could be credited, even if it’s just including at least one of the words from the topic of the essay, and move briskly on.

Larry Wright’s Motley for the 6th is a riff on the idea tips are impossibly complicated to calculate. And that any mathematics might as well be algebra. My question: what the heck calculation is Debbie describing here? There are different ways to find a 15 percent tip. One two-step one is to divide the bill by ten, which is easy and gets you 10 percent. Then divide that by two, which is not-hard, and gets you 5 percent. Add together the 10 percent and 5 percent and you get 15 percent. A one-step method is to just divide by six. This gets you a bit under 17 percent, but that’s close enough. It just requires an ability to divide by six.

There’s other ways to go about it, surely. There are many ways to do any calculation you like. Some of them have the advantages of requiring fewer steps. Some require more steps, but hopefully easier steps. Debbie is, obviously, just describing a nonsensically complicated calculation, to fit the needs of the joke. I’m just trying to think of what a plausible process would lead into the first panel and still get the right answer.

My many Reading the Comics posts are at this link. Essays which mention Sherman’s Lagoon should be at this link. Other essays with The Buckets should appear at this link. And other essays discussing Motley Classics should be here.

## Reading the Comics, August 29, 2018: The Week I Missed One Edition

Have you ever wondered how I gather comic strips for these Reading the Comics posts? Sure, why not go along with me. Well, I do it by this: I read a lot of comic strips. When I run across one that’s got some mathematical theme, I copy the URL for it over to a page of notes. Then I go back to those notes and write up a paragraph or so on each. That is, I do it exactly the way you might imagine if you weren’t very imaginative or trying hard. I explain all this to say that I made a note that I then didn’t notice. So I missed a comic strip. And opened myself up to wondering if there’s an etymological link between “note” and “notice”. Anyway, it’s here. I’m just explaining why it’s late.

Jim Toomey’s Sherman’s Lagoon for the 19th of August is the belated inclusion. It’s a probability strip. It’s built partly on how badly people estimate probability, especially of rare events. And of how badly people estimate the consequences of rare events. For anything that isn’t common, our feelings about the likelihood of events are garbage. And even for common events we’re not that good.

But then it’s hard to quantify a low-probability event too. Take the claim that a human has one chance in 3.7 million of being attacked by a shark. We’ll pretend that’s the real number; I don’t know what is. (I’m suspicious of the ‘3-7’. People picking a random two-digit number are surprisingly likely to pick 37 because, I guess, it ‘feels’ random.) Is that over their lifetime? Over a summer? In a single swimming event? In any case it’s such a tiny chance it’s not worth serious worry. But even then, a person who lives in Wisconsin and only ever swims in Lake Michigan has a considerably smaller chance of shark attack than a person from New Jersey who swims at the Shore. At least some of these things are probabilities we can affect.

So the fellow may be irrational, denying himself something he’d enjoy based on a fantastically unlikely event. But he is acting to avoid something he’s decided he doesn’t want to risk. And, you know, we all act irrationally at times, or else I couldn’t justify buying a lottery ticket every eight months or so. Also is Fillmore (the turtle) the person who needs to hear this argument?

Gary McCoy and Glenn McCoy’s The Duplex for the 26th is an accounting joke. And a cry about poverty, with the idea that one could make the adding up of one’s assets and debts work only by making mathematics logically inconsistent. Or maybe inconsistent. Arithmetic modulo a particular number could be said to make zero equal to some other number, after all, and that’s all valid. Useful, too, especially in enciphering messages and in generating random numbers. It’s less useful for accounting, though. At least it would draw attention if used unilaterally.

Steve Kelley and Jeff Parker’s Dustin for the 28th is roughly a student-resisting-the-homework problem. From the first panel I thought Hayden might be complaining that ‘x’ was used, once again, as the variable to be solved for. It is the default choice, made because we all grew up learning of ‘x’ as the first choice for a number with a not-yet-known identity. ‘y’ and ‘z’ come in as second and third choices, most likely because they’re quite close to ‘x’. Sometimes another letter stands out, usually because the problem compels it. If the framing of the problem is about when events happen then ‘t’ becomes the default choice. If the problem suggests circular motion then ‘r’ or ‘θ’ — radius and angle — become compelling. But if we know no context, and have only the one variable, then ‘x’ it is. It seems weird to do otherwise.

Bill Holbrook’s On The Fastrack for the 28th is part of a week of Fi talking about mathematics to kids. She occasionally delivers seminars meant to encourage enthusiasm about mathematics. I love the principle although I don’t know how long the effect lasts. (Although it is kind of what I’m doing here. Except I think maybe Fi gets paid.) Holbrook’s strips of this mode often include nice literal depictions of metaphors. This week didn’t offer much chance for that particular artistic play.

I have at least one, and often several, Reading the Comics posts, each week. They should all appear at this link. Other essays with Sherman’s Lagoon will appear at this link when they’re written. I’m surprised to learn that’s a new tag. Essays that mention The Duplex are at this link. Other appearances by Dustin, a character who does not appear in this particular essay’s strips, are at this link. And On The Fastrack mentions should appear at this link. Thank you.