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.
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.
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.)
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.
I like this scheme where I use the Sunday publication slot to list comics that mention mathematics without inspiring conversation. I may need a better name for that branch of the series, though. But, nevertheless, here are comic strips from last week that don’t need much said about them.
John Deering’s Strange Brew for the 24th features Pythagoras, here being asked about his angles. I’m not aware of anything actually called a Pythagorean Angle, but there’s enough geometric things with Pythagoras’s name attached for the joke to make sense.
Maria Scrivan’s Half Full for the 25th is a Venn Diagram joke for the week. It doesn’t quite make sense as a Venn Diagram, as it’s not clear to me that “invasive questions” is sensibly a part of “food”. But it’s a break from every comic strip doing a week full of jokes about turkeys preferring to not be killed.
Tony Carrillo’s F Minus for the 26th is set in mathematics class. And talks about how the process of teaching mathematics is “an important step on the road to hating math”, which is funny because it’s painfully true.
I thought I had a flood of mathematically-themed comic strips last week. On reflection, many of them were slight enough not to need further context. You’ll see in the paragraph of not-discussed strips at the end of this. What did rate discussion turned out to get more interesting to me the more I wrote about them.
Stephen Beals’s Adult Children for the 6th uses mathematics as icon of things that are indisputably true. Two plus two equals four is a good example of such. If we take the ordinary meanings of ‘two’ and ‘plus’ and ‘equals’ and ‘four’ there’s no disputing it. The result follows from some uncontroversial-seeming axioms and a lot of deduction. By the rules of logic, the conclusion has to be true, whoever makes it. Even, for that matter, if nobody makes it. It’s difficult to imagine a universe in which nobody ever notices two plus two equals four. But we can imagine that there are mathematical truths that will never be noticed by anyone. (Here’s one. There is some largest finite whole number that any human-created project will ever use in any context. Consider the equation represented by “that number plus two equals (even bigger number)”.)
But you see cards palmed there. What do we mean by ‘two’? Have we got a good definition? Might there be a different definition that’s more useful? Probably not, for ‘two’ anyway. But a part of mathematics, especially as a field develops, is working out what are the important concepts, and what their definitions should be. What a ‘function’ is, for example, went through a lot of debate and change over the 19th century. There is an elusiveness to facts, even in mathematics, where you’d think epistemology would be simpler.
Frank Page’s Bob the Squirrel for the 6th continues the SAT prep questions from earlier in the week. There’s two more problems in shuffling around algebraic expressions here. The first one, problem 5, is probably easiest to do by eliminating wrong answers. is a tedious mess. But look at just the terms: they have to add up to , so, the answer has to be either c or d. So next look at the terms and oh, that’s nice. They add up to zero. The answer has to be c. If you feel like checking the terms, go ahead; that’ll offer some reassurance, if you do the addition correctly.
The second one, problem 8, is probably easier to just think out. If then there’s a lot of places to go. What stands out to me is that has the reciprocal of in it. So, the reciprocal of has to equal the reciprocal of . So . And is, well, four times , so, four times one-half, or two. There’s other ways to go about this. In honestly, what I did when I looked at the problem was multiply both sides of by . But it’s harder to explain why that struck me as an obviously right thing to do. It’s got shortcuts I grew into from being comfortable with the more methodical approach. Someone who does a lot of problems like these will discover shortcuts.
Rick Detorie’s One Big Happy for the 6th asks one of those questions you need to be a genius or a child to ponder. Why don’t the numbers eleven and twelve follow the pattern of the other teens, or for that matter of twenty-one and thirty-two, and the like? And the short answer is that they kind of do. At least, “eleven” and “twelve”, etymologists agree, derive from the Proto-Germanic “ainlif” and “twalif”. If you squint your mouth you can get from “ain” to “one” (it’s probably easier if you go through the German “ein” along the way). Getting from “twa” to “two” is less hard. If my understanding is correct, etymologists aren’t fully agreed on the “lif” part. But they are settled on it means the part above ten. Like, “ainlif” would be “one left above ten”. So it parses as one-and-ten, putting it in form with the old London-English preference for one-and-twenty or two-and-thirty as word constructions.
It’s not hard to figure how “twalif” might over centuries mutate to “twelve”. We could ask why “thirteen” didn’t stay something more Old Germanic. My suspicion is that it amounts to just, well, it worked out like that. It worked out the same way in German, which switches to “-zehn” endings from 13 on. Lithuanian has all the teens end with “-lika”; Polish, similarly, but with “-ście”. Spanish — not a Germanic language — has “custom” words for the numbers up to 15, and then switches to “diecis-” as a prefix to the numbers 6 through 9. French doesn’t switch to a systematic pattern until 17. (And no I am not going to talk about France’s 80s and 90s.) My supposition is that different peoples came to different conclusions about whether they needed ten, or twelve, or fifteen, or sixteen, unique names for numbers before they had to resort to systemic names.
Doug Savage’s Savage Chickens for the 6th is a percentages comic. It makes reference to an old series of (American, at least) advertisements in which four out of five dentists would agree that chewing sugarless gum is a good thing. Shifting the four-out-of-five into 80% riffs is not just fun with tautologies. Percentages have this connotation of technical precision; 80% sounds like a more rigorously known number than “four out of five”. It doesn’t sound as scientific as “0.80”, quite. But when applied to populations a percentage seems less bizarre than a decimal.
This week was another average-grade week of mathematically-themed comic strips. I wonder if I should track them and see what spurious correlations between events and strips turn up. That seems like too much work and there’s better things I could do with my time, so it’s probably just a few weeks before I start doing that.
Ruben Bolling’s Super-Fun-Pax Comics for the 19th is an installment of A Voice From Another Dimension. It’s in that long line of mathematics jokes that are riffs on Flatland, and how we might try to imagine spaces other than ours. They’re taxing things. We can understand some of the rules of them perfectly well. Does that mean we can visualize them? Understand them? I’m not sure, and I don’t know a way to prove whether someone does or does not. This wasn’t one of the strips I was thinking of when I tossed “shapes” into the edition title, but you know what? It’s close enough to matching.
Olivia Walch’s Imogen Quest for the 20th — and I haven’t looked, but it feels to me like I’m always featuring Imogen Quest lately — riffs on the Monty Hall Problem. The problem is based on a game never actually played on Monty Hall’s Let’s Make A Deal, but very like ones they do. There’s many kinds of games there, but most of them amount to the contestant making a choice, and then being asked to second-guess the choice. In this case, pick a door and then second-guess whether to switch to another door. The Monty Hall Problem is a great one for Internet commenters to argue about while the rest of us do something productive. The trouble — well, one trouble — is that whether switching improves your chance to win the car is that whether it does depends on the rules of the game. It’s not stated, for example, whether the host must open a door showing a goat behind it. It’s not stated that the host certainly knows which doors have goats and so chooses one of those. It’s not certain the contestant even wants a car when, hey, goats. What assumptions you make about these issues affects the outcome.
If you take the assumptions that I would, given the problem — the host knows which door the car’s behind, and always offers the choice to switch, and the contestant would rather have a car, and such — then Walch’s analysis is spot on.
Todd Clark’s Lola for the 20th does the old “when do I need to know algebra” question and I admit getting grumpy like this when people ask. Do French teachers have to put up with this stuff?
Brian Fies’s Mom’s Cancer rerun for the 23rd is from one of the delicate moments in her story. Fies’s mother just learned the average survival rate for her cancer treatment is about five percent and, after months of things getting haltingly better, is shaken. But as with most real-world probability questions context matters. The five-percent chance is, as described, the chance someone who’d just been diagnosed in the state she’d been diagnosed in would survive. The information that she’s already survived months of radiation and chemical treatment and physical therapy means they’re now looking at a different question. What is the chance she will survive, given that she has survived this far with this care?
Mark Anderson’s Andertoons for the 24th is the Mark Anderson’s Andertoons for the week. It’s a protesting-student kind of joke. For the student’s question, I’m not sure how many sides a polygon has before we can stop memorizing them. I’d say probably eight. Maybe ten. Of the shapes whose names people actually care about, mm. Circle, triangle, a bunch of quadrilaterals, pentagons, hexagons, octagons, maybe decagon and dodecagon. No, I’ve never met anyone who cared about nonagons. I think we could drop heptagons without anyone noticing either. Among quadrilaterals, ugh, let’s see. Square, rectangle, rhombus, parallelogram, trapezoid (or trapezium), and I guess diamond although I’m not sure what that gets you that rhombus doesn’t already. Toss in circles, ellipses, and ovals, and I think that’s all the shapes whose names you use.