One is that, as we’ve thought of counting numbers, there is always “one more”. This doesn’t have to be. We could work with perfectly good number systems that have a largest number. We do, in fact. Every computer programming language has some largest integer that it will deal with. If you need a larger number, you have to do something clever. Your clever idea will let you address some range of bigger numbers, but it too will have a maximum. We’ve set those limits large enough that, usually, they’re not an inconvenience. They’re still there.
But those limits are forced on us by the many failings of matter. What when we get just past Plato’s line’s division, into the reasoning of pure mathematics? There we can set up counting numbers. The standard way to do this is to suppose there is a number “1”. And to suppose that, for any counting number we have, there is a successor, a number one-plus-that. If Joey were to ask why there has to be, all Dennis could do is shrug. This makes an axiom out of there always being one more. If you don’t like it, make some other arithmetic. Anyway we only understand any of this using fallible matter, so good luck.
This progression can be heady, though. The counting numbers are probably the most understandable infinitely large set there is. Thinking about them seriously can induce the sort of dizzy awe that pondering Deep Time or the vastness of space can do. That seems a bit above Dennis’s age level, but some people are stricken with the infinite sooner than others are.
Charles Schulz’s Peanuts Begins rerun for the 2nd has Charlie Brown dismiss arithmetic as impractical. It fits the motif of mathematics as an unworldly subject. There’s the common joke that pure mathematics even dreams of being of no use to anyone. Arithmetic, though, has always been a practical subject. It introduces us to many abstract ideas, particularly group theory. This subject looks at what we can do with systems that work like arithmetic without necessarily having numbers, or anything that works with numbers.
And a couple of comic strips mentioned mathematics, although in too slight a way to discuss. Dana Simpson’s Phoebe and her Unicorn on the 30th of April started a sequence in which doodles on Phoebe’s homework came to life. That it’s mathematics homework was mostly incidental. I’m open to the argument that mathematics encourages doodling in a way that, say, spelling does not. I’d also be open to the argument you aren’t doing geometry if you don’t doodle. Anyway. Dan Thompson’s Brevity for the 2nd of May features Sesame Street’s Count von Count. It’s a bit of wordplay on the use of “numbers” for songs. And, of course, the folkloric tradition of vampires as compulsive counters.
There’s several comics from the first half of last week that I can’t perfectly characterize. They seem to be on-topic enough for my mathematical discussions. It’s just how exactly they are on-topic that I haven’t quite got. Some weeks are like that.
Dave Whamond’s Reality Check for the 28th circles around being a numerals joke. It’s built on the binary representation of numbers that we’ve built modern computers on. And on the convention that “(Subject) 101” is the name for an introductory course in a subject. This convention of course numbering — particularly, three-digit course numbers, with the leading digit representing the year students are expected to take it — seems to have spread in American colleges in the 1930s. It’s a compromise, as many things are. As college programs of study become more specialized there’s the need for a greater number of courses in each field. And there’s a need to give people some hint of the course level. “Numerical Methods” could be a sophomore, senior, or grad-student course; how should someone from a different school know what to expect? But the pull of the serial number, and the idea that ’01’ must be the first in a field, is hard to resist.
Anyway, the long string of zeroes and ones after the original ‘101’ is silliness and that’s all it has to be. The number one-hundred-and-one in binary would be a mere “1100101”, which doesn’t start with the important one-oh-one, and isn’t a big enough string of digits to be funny. Maybe this is a graduate course. The number given, if we read it as a single long binary number, would be 182,983,026,468. I’ve been to schools which use four-digit course codes. Twelve digits seems excessive.
John Deering’s Strange Brew for the 29th circles around being an anthropomorphic numerals joke. At least it is a person using a large representation of the number eight. I’m not sure how to characterize it, or why I find the strip amusing. It’s a strange one.
Thaves’s Frank and Ernest for the 1st is, finally, a certain anthropomorphic numerals joke. With wordplay about prime numbers being unavoidably prime suspects. … And when I was a kid, I had no idea what “numbers rackets” were, other than a thing sometimes mentioned on older sitcoms. That it involved somehow literally taking numbers and doing … something … that the authorities didn’t like was mysterious. I don’t remember what surely hilarious idea the young me had for what that might even mean. I suspect that, had I seen this strip at the time, I would have understood this wasn’t really whatever was going on. But I would have explained to my parents what a prime number was, and they would put up with my doing so, because that’s just what our relationship was.
Dave Whamond’s Reality Check for the 1st is more or less the Venn Diagram joke for this essay. It’s a bit of a fourth-wall-breaking strip: the joke wouldn’t really work from the other goldfish’s perspective. Anyway, only two of those figures are proper Venn diagrams. The topmost figure, with five circles, and the bottommost, with three, aren’t proper Venn diagrams. Only some of the possible intersections between sets exist there. They are proper Euler diagrams, though.
And we had another of those peculiar days where a lot of strips are on-topic enough for me to talk about.
Eric the Circle, this one by Kyle, for the 26th has a bit of mathematical physics in it. This is the kind of diagram you’ll see all the time, at least if you do the mathematics that tells you where things will be and when. The particular example is an easy problem, a thing rolling down an inclined plane. But the work done for it applies to more complicated problems. The question it’s for is, “what happens when this thing slides down the plane?” And that depends on the forces at work. There’s gravity, certainly . If there were something else it’d be labelled. Gravity’s represented with that arrow pointing straight down. That gives us the direction. The label (Eric)(g) gives us how strong this force is.
Where the diagram gets interesting, and useful, are those dashed lines ending in arrows. One of those lines is, or at least means to be, parallel to the incline. The other is perpendicular to it. These both reflect gravity. We can represent the force of gravity as a vector. That means, we can represent the force of gravity as the sum of vectors. This is like how we can can write “8” or we can write “3 + 5”, depending on what’s more useful for what we’re doing. (For example, if you wanted to work out “67 + 8”, you might be better off doing “67 + 3 + 5”.) The vector parallel to the plane and the one perpendicular to the plane add up to the original gravity vector.
The force that’s parallel to the plane is the only force that’ll actually accelerate Eric. The force perpendicular to the plane just … keeps it snug against the plane. (Well, it can produce friction. We try not to deal with that in introductory physics because it is so hard. At most we might look at whether there’s enough friction to keep Eric from starting to slide downhill.) The magnitude of the force parallel to the plane, and perpendicular to the plane, are easy enough to work out. These two forces and the original gravity can be put together into a little right triangle. It’s the same shape but different size to the right triangle made by the inclined plane plus a horizontal and a vertical axis. So that’s how the diagram knows the parallel force is the original gravity times the sine of x. And that the perpendicular force is the original gravity times the cosine of x.
The perpendicular force is often called the “normal” force. This because mathematical physicists noticed we had only 2,038 other, unrelated, things called “normal”.
Rick Detorie’s One Big Happy for the 26th sees Ruthie demand to know who this Venn person was. Fair question. Mathematics often gets presented as these things that just are. That someone first thought about these things gets forgotten.
John Venn, who lived from 1834 to 1923 — he died the 4th of April, it happens — was an English mathematician and philosopher and logician and (Anglican) priest. This is not a rare combination of professions. From 1862 he was a lecturer in Moral Science at Cambridge. This included work in logic, yes. But he also worked on probability questions. Wikipedia credits his 1866 Logic Of Chance with advancing the frequentist interpretation of probability. This is one of the major schools of thought about what the “probability of an event” is. It’s the one where you list all the things that could possibly happen, and consider how many of those are the thing you’re interested in. So, when you do a problem like “what’s the probability of rolling two six-sided dice and getting a total of four”? You’re doing a frequentist probability problem.
Venn Diagrams he presented to the world around 1880. These show the relationships between different sets. And the relationships of mathematical logic problems they represent. Venn, if my sources aren’t fibbing, didn’t take these diagrams to be a new invention of his own. He wrote of them as “Euler diagrams”. Venn diagrams, properly, need to show all the possible intersections of all the sets in play. You just mark in some way the intersections that happen to have nothing in them. Euler diagrams don’t require this overlapping. The name “Venn diagram” got attached to these pictures in the early 20th century. Euler here is Leonhard Euler, who created every symbol and notation mathematicians use for everything, and who has a different “Euler’s Theorem” that’s foundational to every field of mathematics, including the ones we don’t yet know exist. I exaggerate by 0.04 percent here.
Venn had his other projects, too. His biography at St Andrews writes of his composing The Biographical History of Gonville and Caius College (Cambridge). And then he had another history of the whole Cambridge University. It also mentions his skills in building machines, though only cites one, a device for bowling cricket balls. The St Andrews biography says that in 1909 “Venn’s machine clean bowled one of [the Australian Cricket Team’s] top stars four times”. I do not know precisely what it means but I infer it to be a pretty good showing for the machine. His Wikipedia biography calls him a “passionate gardener”. Apparently the Cambridgeshire Horticultural Society awarded him prizes for his roses in July 1885 and for white carrots in September that year. And that he was a supporter of votes for women.
Ashleigh Brilliant’s Pot-Shots for the 26th makes a cute and true claim about percentiles. That a person will usually be in the upper 99% of whatever’s being measured? Hard to dispute. But, measure enough things and eventually you’ll fall out of at least one of them. How many things? This is easy to calculate if we look at different things that are independent of each other. In that case we could look at 69 things before there we’d expect a 50% chance of at least one not being in the upper 99%.
It’s getting that independence that’s hard. There’s often links between things. For example, a person’s height does not tell us much about their weight. But it does tell us something. A person six foot, ten inches tall is almost certainly not also 35 pounds, even though a person could be that size or could be that weight. A person’s scores on a reading comprehension test and their income? But test-taking results and wealth are certainly tied together. Age and income? Most of us have a bigger income at 46 than at 6. This is part of what makes studying populations so hard.
T Shepherd’s Snow Sez for the 26th is finally a strip I can talk about briefly, for a change. Snow does a bit of arithmetic wordplay, toying with what an expression like “1 + 1” might represent.
Last week started out at a good clip: two comics with enough of a mathematical theme I could imagine writing a paragraph about them each day. Then things puttered out. The rest of the week had almost nothing. At least nothing that seemed significant enough. I’ll list those, since that’s become my habit, at the end of the essay.
Perhaps it’s so. Some societies have been found to have, what seem to us, rather few numerals. This doesn’t reflect on anyone’s abilities or intelligence or the like. And it doesn’t mean people who lack a word for, say, “forty-nine” would be unable to compute. It might take longer, but probably just from inexperience. If someone practiced much calculation on “forty-nine” they’d probably have a name for it. And folks raised in the western mathematics use, even enjoy, some vagueness about big numbers too. We might say there are “dozens” of a thing even if there are not precisely 24, 36, or 48 of the thing; “52” is close enough and we probably didn’t even count it up. “Hundred” similarly has gotten the connotation of being a precise number, but it’s used to mean “really quite a lot of a thing”. The words “thousands”, “millions”, and mock-numbers like “zillions” have a similar role. They suggest different ranges of what might be “many”.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th is a SABRmetrics joke! At least, it’s an optimization joke, built on the idea that you can find an optimum strategy for anything, whether winning baseball games or The War. The principle is hard to argue with. Nobody would doubt that different approaches to a battle affect how likely winning is. We can imagine gathering data on how different tactics affect the outcome. (We can easily imagine combat simulators running these experiments, particularly.)
The catch — well, one catch — is that this tempts one to reward a process. Once it’s taken for granted the process works, then whether it’s actually doing what you want gets forgotten. And once everyone knows what’s being measured it becomes possible to game the system. Famously, in the mid-1960s the United States tried to judge its progress in the Vietname War by counting the number of enemy soldiers killed. There was then little reason to care about who was killed, or why. And reason to not care whether actual enemy soldiers were being killed. There’s good to be said about testing whether the things you try to do work. There’s great danger in thinking that the thing you can measure guarantees success.
Mark Anderson’s Andertoons for the 21st is a bit of fun with definitions. Mathematicians rely on definitions. It’s hard to imagine a proof about something undefined. But definitions are hard to compose. We usually construct a definition because we want a common term to describe a collection of things, and to exclude another collection of things. And we need people like Wavehead who can find edge cases, things that seem to satisfy a definition while breaking its spirit. This can let us find unstated assumptions that we should pay attention to. Or force us to accept that the definition is so generally useful that we’ll tolerate it having some counter-intuitive implications.
My favorite counter-intuitive implication is in analysis. The field has a definition for what it means that a function is continuous. It’s meant to capture the idea that you could draw a curve representing the function without having to lift the pen that does it. The best definition mathematicians have settled on allows you to count a function that’s continuous at a single point in all of space. Continuity seems like something that should need an interval to happen. But we haven’t found a better way to define “continuous” that excludes this pathological case. So we embrace the weirdness in exchange for general usefulness.
Charles Brubaker’s Ask A Cat for the 21st is a guest appearance from Brubaker’s other strip, The Fuzzy Princess. It’s a rerun and I did discuss it earlier. Soap bubbles make for great mathematics. They’re easy to play with, for one thing. That’s good for capturing imagination. And the mathematics behind them is deep, and led to important results analytically and computationally. It happens when this strip first ran I’d encountered a triplet of essays about the mathematics of soap bubbles and wireframe surfaces. My introduction to those essays is here.
Benita Epstein’s Six Chix for the 25th I wasn’t sure I’d include. But Roy Kassinger asked about it, so that tipped the scales. The dog tries to blame his bad behavior on “the algorithm”, bringing up one of the better monsters of the last couple years. An algorithm is just the procedure by which you do something. Mathematically, that’s usually to solve a problem. That might be finding some interesting part of the domain or range of a function. That might be putting a collection of things in order. that might be any of a host of things. And then we go make a decision based on the results of the algorithm.
What earns The Algorithm its deserved bad name is mindlessness. The idea that once you have an algorithm that a problem is solved. Worse, that once an algorithm is in place it would be irrational to challenge it. I have seen the process termed “mathwashing”, by analogy with whitewashing, and it’s a good one. The notion that because something is done by computer it must be done correctly is absurd. We knew it was absurd before there were computers as we knew them, as see anyone for the past century who has spoken of a “Kafkaesque” interaction with a large organization. It’s impossible to foresee all the outcomes of any reasonably complicated process, much less to verify that all the outcomes are handled correctly. This is before we consider that there will always be mistakes made in the handling of data. Or in the carrying out of the process. And that’s before we consider bad actors. I’m sure there must be research into algorithms designed to handle gaming of the system. I don’t know that there are any good results yet, though. We certainly need them.
After that busy start last Sunday, Comic Strip Master Command left only a few things for the rest of the week. Here’s everything that seemed worthy of some comment to me:
Alex Hallatt’s Arctic Circle for the 12th is an arithmetic cameo. It’s used as the sort of thing that can be tested, with the straightforward joke about animal testing to follow. It’s not a surprise that machines should be able to do arithmetic. We’ve built machines for centuries to do arithmetic. Literally; Wilhelm Gottfried Leibniz designed and built a calculating machine able to add, subtract, multiply, and divide. This accomplishment from one of the founders of integral calculus is a potent reminder of how much we can accomplish if we’re supposed to be writing instead. (That link is to Robert Benchley’s classic essay “How To Get Things Done”. It is well worth reading, both because it is funny and because it’s actually good, useful advice.)
But it’s also true that animals do know arithmetic. At least a bit. Not — so far as we know — to the point they ponder square roots and such. But certainly to count, to understand addition and subtraction roughly, to have some instinct for calculations. Stanislas Dehaene’s The Number Sense: How the Mind Creates Mathematics is a fascinating book about this. I’m only wary about going deeper into the topic since I don’t know a second (and, better, third) pop book touching on how animals understand mathematics. I feel more comfortable with anything if I’ve encountered it from several different authors. Anyway it does imply the possibility of testing a polar bear’s abilities at arithmetic, only in the real world.
Berkeley Breathed’s Bloom County rerun for the 13th has another mathematics cameo. Geometry’s a subject worthy of stoking Binkley’s anxieties, though. It has a lot of definitions that have to be carefully observed. And while geometry reflects the understanding we have of things from moving around in space, it demands a precision that we don’t really have an instinct for. It’s a lot to worry about.
Terry Border’s Bent Objects for the 15th is our Venn Diagram joke for the week. I like this better than I think the joke deserves, probably because it is done in real materials. (Which is the Bent Objects schtick; it’s always photographs of objects arranged to make the joke.)
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 15th is a joke on knowing how far to travel but not what direction. Normal human conversations carry contextually reasonable suppositions. Told something is two miles away, it’s probably along the major road you’re on, or immediately nearby. I’d still ask for clarification told something was “two miles away”. Two blocks, I’d let slide, on the grounds that it’s no big deal to correct a mistake.
Still, mathematicians carry defaults with them too. They might be open to a weird, general case, certainly. But we have expectations. There’s usually some obvious preferred coordinate system, or directions. If it’s important that we be ready for alternatives we highlight that. We specify the coordinate system we want. Perhaps we specify we’re taking that choice “without loss of generality”, that is, without supposing some other choice would be wrong.
I noticed the mathematician’s customized plate too. “EIPI1” is surely a reference to the expression . That sum, it turns out, equals zero. It reflects this curious connection between exponentiation, complex-valued numbers, and the trigonometric functions. It’s a weird thing to know is true, and it’s highly regarded in certain nerd circles for that weirdness.
Through the end of December my Fall 2018 Mathematics A To Z continues. I’m still open for topics to discuss from the last half-dozen letters of the alphabet. Even if someone’s already given a word for some letter, suggest something anyway. You might inspire me in good ways.
I’d like to add something to my roundup up of last week’s mathematically-themed comic strips. That thing is a reminder that I’m hosting this month’s Playful Mathematics Education Carnival. It’ll post the last week of September. If you’ve recently seen pages that teach, that play games, that show any kind of mathematics that makes you smile, please, let me know. It’s worth sharing with more people.
Tom Gammill’s The Doozies for the 6th is the Venn Diagrams joke for the week. It’s only a two-circle diagram, but the comic strip hasn’t got that large a cast. And, really, would be hard to stage in a way that made the joke communicable with three or four participants.
Is that superhuman? Well, obviously, literally not. But it’s beyond what most of us could imagine doing. I admit I can’t imagine keeping anything straight in my head for nine hours. But. The basic rules of addition aren’t that exotic. Even a process like finding square roots can be done as additions and divisions and multiplications. Much of what makes this look hard is memory. How do you keep track of a hundred or so partial results each of a hundred or so digits? Much of what else is hard is persistence. How do you keep going after the seventh hour of this? And both are traits that you can develop, and practice, and at least get a little better on.
Or bypass the hard work. If asked 235 plus 747 I’d at least answer “a bit under a thousand”, which isn’t bad for an instant answer. 235 is a little under 250; 747 a little under 750; and 250 plus 750 is easy. Rewrite 235 as 250 – 15, and 747 as 750 – 3, and you have this: 235 + 747 is 250 + 750 – 15 – 3. So that’s 1000 minus 18. 982, pretty attainable. This takes practice. It amounts to learning how to spot an easy problem that looks like the question you actually have.
Greg Evans’s Luann Againn for the 7th shows a date living up to its potential as a fiasco. But it’s not a surprise Gunther finds himself comfortable talking trigonometry. The subject is not one that most people find cozy. I’d guess most people on introduction see it as some weird hybrid that fuses the impenetrable diagrams of geometry with the baffling formulas of algebra.
But there’s comfort in it, especially to a particular personality type. There are a lot of obscure things making up trigonometry. But there’s this beauty, too. All the basic trigonometry functions are tied together in neat little pairs and triplets. Formulas connect the properties of an angle with those of its half and its double. There’s a great many identities, particular calculations that have the same value for every angle.
You can say that about anything, of course. Any topic humans study has endless fascination. What makes mathematical fields comfortable? For one, that they promise this certain knowability. Trigonometry has a jillion definitions and rules and identities and all that. But that means you have a great many things of absolute reliability. They offer this certainty that even “hard” sciences like physics don’t have. Far more security than you see with the difficult sciences, like biology or sociology. And true dependability, compared to the mystifying and obscure rules of interacting with other humans. If you don’t feel you know how to be with people, and don’t feel like you could ever learn, a cosecant is at least something you can master.
Although the hyperbolic cosine is interesting and I could go on about it.
Eric the Circle for the 18th of June is a bit of geometric wordplay for the week. A secant is — well, many things. One of the important things is it’s a line that cuts across a circle. It intersects the circle in two points. This is as opposed to a tangent, which touch it in one. Or missing it altogether, which I think hasn’t got any special name. “Secant” also appears as one of the six common trig functions out there.
In value the secant of an angle is just the reciprocal of the cosine of that angle. Where the cosine is never smaller than -1 nor larger than 1, the secant is always either greater than 1 or smaller than -1. It’s a useful function to have by name. We can write “the secant of angle θ” as . The otherwise sensible-looking is unavailable, because we use that to mean “the angle whose cosine is θ”. We need to express that idea, the “arc-cosine” or “inverse cosine”, quite a bit too. And would look like we wanted the cosine of one divided by θ. Ultimately, we have a lot of ideas we’d like to write down, and only so many convenient quick shorthand ways to write them. And by using secant as its own function we can let the arc-cosine have a convenient shorthand symbol. These symbols are a point where you see the messy, human, evolutionary nature of mathematical symbols at work.
We can understand the cosine of an angle θ by imagining a right triangle with hypotenuse of length 1. Set that so the hypotenuse makes angle θ with respect to the x-axis. Then the opposite leg of that right triangle will be the cosine of θ away from the origin. The secant, now, that works differently. Again here imagine a right triangle, but this time one of the legs has length 1. And that leg is at an angle θ with respect to the x-axis. Then the far leg of that right triangle is going to cross the x-axis. And it’ll do that at a point that’s the secant of θ away from the origin.
Larry Wright’s Motley Classics for the 19th speaks of algebra as the way to explain any sufficiently complicated thing. Algebra’s probably not the right tool to analyze a soap opera, or any drama really. The interactions of characters are probably more a matter for graph theory. That’s the field that studies groups of things and the links between them. Occasionally you’ll see analyses of, say, which characters on some complicated science fiction show spend time with each other and which ones don’t. I’m not aware of any that were done on soap operas proper. I suspect most mathematics-oriented nerds view the soaps as beneath their study. But most soap operas do produce a lot of show to watch, and to summarize; I can’t blame them for taking a smaller, easier-to-summarize data set to study. (Also I’m not sure any of these graphs reveal anything more enlightening than, “Huh, really thought The Doctor met Winston Churchill more often than that”.)
Dave Whamond’s Reality Check for the 19th is our Venn Diagram strip for the week. I say the real punch line is the squirrel’s, though. Properly, yes, the Venn Diagram with the two having nothing in common should still have them overlap in space. There should be a signifier inside that there’s nothing in common, such as the null symbol or an x’d out intersection. But not overlapping at all is so commonly used that it might as well be standard.
Teresa Bullitt’s Frog Applause for the 21st uses a thought balloon full of mathematical symbols as icon for far too much deep thinking to understand. I would like to give my opinion about the meaningfulness of the expressions. But they’re too small for me to make out, and GoComics doesn’t allow for zooming in on their comics anymore. I looks like it’s drawn from some real problem, based on the orderliness of it all. But I have no good reason to believe that.
Hi, all. I apologize for being late in posting this, but my Friday and Saturday were eaten up by pinball competition. Pinball At The Zoo, particularly, in Kalamazoo, Michigan. There, Friday, I stepped up first thing and put in four games on the Classics, pre-1985, tournament bank and based on my entry scores was ranked the second-best player there. And then over the day my scores dwindled lower and lower on the list of what people had entered until, in the last five minutes of qualifying, they dropped off the roster altogether and I was knocked out. Meanwhile in the main tournament, I was never even close to making playoffs. But I did have a fantastic game of Bally/Midway’s World Cup Soccer, a game based on how much the United States went crazy for soccer that time we hosted the World Cup for some reason. The game was interrupted by one of the rubber straps around one of the kickers (the little triangular table just past the flippers that you would think would be called the bumpers) breaking, and then by the drain breaking in a way that later knocked the game entirely out of the competition. So anyway besides that glory I’ve been very busy trying to figure out what’s gone wrong and stepping outside to berate the fox squirrels out back, and that’s why I’m late with all this. I’m sure you relate.
Bill Holbrook’s Kevin and Kell rerun for the 15th is the anthropomorphic numerals strip for the week. Also the first of the anthropomorphic strips for the week. Calculating taxes has always been one of the compelling social needs for mathematics, arithmetic especially. If we consider the topic to be “accounting” then that might be the biggest use of mathematics in society. At least by humans; I’m not sure how to rate the arithmetic that computers do even for not explicitly mathematical tasks like sending messages back and forth. New comic strip tag for around here, too.
Bill Schorr’s The Grizzwells for the 17th sees Fauna not liking trigonometry class. I’m sympathetic. I remember it as seeming to be a lot of strange new definitions put to vague purposes. On the bright side, when you get into calculus trigonometry starts solving more problems than it creates. On the dim side, at least when I took it they tried to pass off “trigonometric substitution” as a thing we might need. (OK, it’s come in useful sometimes, but not as often as the presentation made it look.) Also a new comic strip tag.
Eric the Circle for the 18th, this one by sdhardie, is a joke in the Venn Diagram mode. The strip’s a little unusual for not having one of the circles be named Eric. Not a new comic strip tag.
Lord Birthday’s Dumbwitch Castle for the 19th is a small sketch and mostly a list of jokes. This is the normal format for this strip, which tests the idea of what makes something a comic strip. I grant it’s a marginal inclusion, but I am tickled by the idea of a math slap so here you go. This one’s another new comic strip tag.
The back half of last week’s mathematically themed comic strips aren’t all that deep. They make up for it by being numerous. This is how calculus works, so, good job, Comic Strip Master Command. Here’s what I have for you.
Mark Anderson’s Andertoons for the 20th marks its long-awaited return to these Reading The Comics posts. It’s of the traditional form of the student misunderstanding the teacher’s explanations. Arithmetic edition.
Marty Links’s Emmy Lou for the 20th was a rerun from the 22nd of September, 1976. It’s just a name-drop. It’s not like it matters for the joke which textbook was lost. I just include it because, what the heck, might as well.
Jef Mallett’s Frazz for the 21st uses the form of a story problem. It’s a trick question anyway; there’s really no way the Doppler effect is going to make an ice cream truck’s song unrecognizable, not even at highway speeds. Too distant to hear, that’s a possibility. Also I don’t know how strictly regional this is but the ice cream trucks around here have gone in for interrupting the music every couple seconds with some comical sound effect, like a “boing” or something. I don’t know what this hopes to achieve besides altering the timeline of when the ice cream seller goes mad.
Mark Litzler’s Joe Vanilla for the 21st I already snuck in here last week, in talking about ‘x’. The variable does seem like a good starting point. And, yeah, hypothesis block is kind of a thing. There’s nothing quite like staring at a problem that should be interesting and having no idea where to start. This happens even beyond grade school and the story problems you do then. What to do about it? There’s never one thing. Study it a good while, read about related problems a while. Maybe work on something that seems less obscure a while. It’s very much like writer’s block.
Ryan North’s Dinosaur Comics rerun for the 22nd straddles the borders between mathematics, economics, and psychology. It’s a problem about making forecasts about other people’s behavior. It’s a mystery of game theory. I don’t know a proper analysis for this game. I expect it depends on how many rounds you get to play: if you have a sense of what people typically do, you can make a good guess of what they will do. If everyone gets a single shot to play, all kinds of crazy things might happen.
Jef Mallet’s Frazz gets in again on the 22nd with some mathematics gibberish-talk, including some tossing around of the commutative property. Among other mistakes Caulfield was making here, going from “less is more to therefore more is less” isn’t commutation. Commutation is about binary operations, where you match a pair of things to a single thing. The operation commutes if it never matters what the order of the pair of things is. It doesn’t commute if it ever matters, even a single time, what the order is. Commutativity gets introduced in arithmetic where there are some good examples of the thing. Addition and multiplication commute. Subtraction and division don’t. From there it gets forgotten until maybe eventually it turns up in matrix multiplication, which doesn’t commute. And then it gets forgotten once more until maybe group theory. There, whether operations commute or not is as important a divide as the one between vertebrates and invertebrates. But I understand kids not getting why they should care about commuting. Early on it seems like a longwinded way to say what’s obvious about addition.
Bud Blake’s Tiger rerun for the 23rd starts with a real-world example of your classic story problem. I like the joke in it, and I also like Hugo’s look of betrayal and anger in the second panel. A spot of expressive art will do so good for a joke.
If there was a theme this week, it was puzzles. So many strips had little puzzles to work out. You’ll see. Thank you.
Bill Amend’s FoxTrot for the 30th of April tries to address my loss of Jumble panels. Thank you, whoever at Comic Strip Master Command passed along word of my troubles. I won’t spoil your fun. As sometimes happens with a Jumble you can work out the joke punchline without doing any of the earlier ones. 64 in binary would be written 1000000. And from this you know what fits in all the circles of the unscrambled numbers. This reduces a lot of the scrambling you have to do: just test whether 341 or 431 is a prime number. Check whether 8802, 8208, or 2808 is divisible by 117. The integer cubed you just have to keep trying possibilities. But only one combination is the cube of an integer. The factorial of 12, just, ugh. At least the circles let you know you’ve done your calculations right.
Steve McGarry’s activity feature Kidtown for the 30th plays with numbers some. And a puzzle that’ll let you check how well you can recognize multiples of four that are somewhere near one another. You can use diagonals too; that’s important to remember.
Mac King and Bill King’s Magic in a Minute feature for the 30th is also a celebration of numerals. Enjoy the brain teaser about why the encoding makes sense. I don’t believe the hype about NASA engineers needing days to solve a puzzle kids got in minutes. But if it’s believable, is it really hype?
Marty Links’s Emmy Lou from the 29th of October, 1963 was rerun the 2nd of May. It’s a reminder that mathematics teachers of the early 60s also needed something to tape to their doors.
Mark Litzler’s Joe Vanilla for the 2nd name-drops the Null Hypothesis. I’m not sure what Litzler is going for exactly. The Null Hypothesis, though, comes to us from statistics and from inference testing. It turns up everywhere when we sample stuff. It turns up in medicine, in manufacturing, in psychology, in economics. Everywhere we might see something too complicated to run the sorts of unambiguous and highly repeatable tests that physics and chemistry can do — things that are about immediately practical questions — we get to testing inferences. What we want to know is, is this data set something that could plausibly happen by chance? Or is it too far out of the ordinary to be mere luck? The Null Hypothesis is the explanation that nothing’s going on. If your sample is weird in some way, well, everything is weird. What’s special about your sample? You hope to find data that will let you reject the Null Hypothesis, showing that the data you have is so extreme it just can’t plausibly be chance. Or to conclude that you fail to reject the Null Hypothesis, showing that the data is not so extreme that it couldn’t be chance. We don’t accept the Null Hypothesis. We just allow that more data might come in sometime later.
I don’t know what Litzler is going for with this. I feel like I’m missing a reference and I’ll defer to a finance blogger’s Reading the Comics post.
I’d been splitting Reading the Comics posts between Sunday and Thursday to better space them out. But I’ve got something prepared that I want to post Thursday, so I’ll bump this up. Also I had it ready to go anyway so don’t gain anything putting it off another two days.
Bill Amend’s FoxTrot Classics for the 27th reruns the strip for the 4th of May, 2006. It’s another probability problem, in its way. Assume Jason is honest in reporting whether Paige has picked his number correctly. Assume that Jason picked a whole number. (This is, I think, the weakest assumption. I know Jason Fox’s type and he’s just the sort who’d pick an obscure transcendental number. They’re all obscure after π and e.) Assume that Jason is equally likely to pick any of the whole numbers from 1 to one billion. Then, knowing nothing about what numbers Jason is likely to pick, Paige would have one chance in a billion of picking his number too. Might as well call it certainty that she’ll pay a dollar to play the game. How much would she have to get, in case of getting the number right, to come out even or ahead? … And now we know why Paige is still getting help on probability problems in the 2017 strips.
Sandra Bell-Lundy’s Between Friends for the 29th also gives me a bit of a break by just being a Venn Diagram-based joke. At least it’s using the shape of a Venn Diagram to deliver the joke. It’s not really got the right content.
Harley Schwadron’s 9 to 5 for the 29th is this week’s joke about arithmetic versus propaganda. It’s a joke we’re never really going to be without again.
I have my reasons for this installment’s title. They involve my deductions from a comic strip. Give me a few paragraphs.
Mark Anderson’s Andertoons for the 16th asks for attention from whatever optician-written blog reads the comics for the eye jokes. And meets both the Venn Diagram and the Mark Anderson’s Andertoons content requirements for this week. Good job! Starts the week off strong.
Lincoln Pierce’s Big Nate: First Class for the 16th, rerunning the strip from 1993, is about impossibly low-probability events. We can read the comic as a joke about extrapolating a sequence from a couple examples. Properly speaking we can’t; any couple of terms can be extended in absolutely any way. But we often suppose a sequence follows some simple pattern, as many real-world things do. I’m going to pretend we can read Jenny’s estimates of the chance she’ll go out with him as at all meaningful. If Jenny’s estimate of the chance she’d go out with Nate rose from one in a trillion to one in a billion over the course of a week, this could be a good thing. If she’s a thousand times more likely each week to date him — if her interest is rising geometrically — this suggests good things for Nate’s ego in three weeks. If she’s only getting 999 trillionths more likely each week — if her interest is rising arithmetically — then Nate has a touch longer to wait before a date becomes likely.
(I forget whether she has agreed to a date in the 24 years since this strip first appeared. He has had some dates with kids in his class, anyway, and some from the next grade too.)
Is the point of the first problem, Farmer Joe’s apples, to see whether a student can do a not-quite-long division? Or is it to see whether the student can extract a price-per-quantity for something, and apply that to find the quantity to fit a given price? If it’s the latter then the numbers don’t make a difference. One would want to avoid marking down a student who knows what to do, and could divide 15 cents by three, but would freeze up if a more plausible price of, say, $2.25 per pound had to be divided by three.
But then the second problem, Mr Schad driving from Belmont to Cadillac, got me wondering. It is about 84 miles between the two Michigan cities (and there is a Reed City along the way). The time it takes to get from one city to another is a fair enough problem. But these numbers don’t make sense. At 55 miles per hour the trip takes an awful 1.5273 hours. Who asks elementary school kids to divide 84 by 55? On purpose? But at the state highway speed limit (for cars) of 70 miles per hour, the travel time is 1.2 hours. 84 divided by 70 is a quite reasonable thing to ask elementary school kids to do.
And then I thought of this: you could say Belmont and Cadillac are about 88 miles apart. Google Maps puts the distance as 86.8 miles, along US 131; but there’s surely some point in the one town that’s exactly 88 miles from some point in the other, just as there’s surely some point exactly 84 miles from some point in the other town. 88 divided by 55 would be another reasonable problem for an elementary school student; 1.6 hours is a reasonable answer. The (let’s call it) 1980s version of the question ought to see the car travel 88 miles at 55 miles per hour. The contemporary version ought to see the car travel 84 miles at 70 miles per hour. No reasonable version would make it 84 miles at 55 miles per hour.
So did Mallett take a story problem that could actually have been on an era-appropriate test and ancient it up?
Before anyone reports me to Comic Strip Master Command let me clarify what I’m wondering about. I don’t care if the details of the joke don’t make perfect sense. They’re jokes, not instruction. All the story problem needs to set up the joke is the obsolete speed limit; everything else is fluff. And I enjoyed working out variation of the problem that did make sense, so I’m happy Mallett gave me that to ponder.
Here’s what I do wonder about. I’m curious if story problems are getting an unfair reputation. I’m not an elementary school teacher, or parent of a kid in school. I would like to know what the story problems look like. Do you, the reader, have recent experience with the stuff farmers, drivers, and people weighing things are doing in these little stories? Are they measuring things that people would plausibly care about today, and using values that make sense for the present day? I’d like to know what the state of story problems is.
John Hambrock’s The Brilliant Mind of Edison Lee for the 18th uses mental arithmetic as the gauge of intelligence. Pretty harsly, too. I wouldn’t have known the square root of 8649 off the top of my head either, although it’s easy to tell that 92 can’t be right: the last digit of 92 squared has to be 4. It’s also easy to tell that 92 has to be about right, though, as 90 times 90 will be about 8100. Given this information, if you knew that 8,649 was a perfect square, you’d be hard-pressed to think of a better guess for its value than 93. But since most whole numbers are not perfect squares, “a little over 90” is the best I’d expect to do.
Comic Strip Master Command sent out a big mass of comics this past week. Today’s installment will only cover about half of them. This half does feature a number of comics that show off jokes that’ve run here before. I’m sure it was coincidence. Comic Strip Master Command must have heard I was considering alerting cartoonists that I was talking about them. That’s fine for something like last week when I could talk about NP-complete problems or why we call something a “hypotenuse”. It can start a conversation. But “here’s a joke treating numerals as if they were beings”? All they can do is agree, that is what the joke is. If they disagree at that point they’re just trying to start a funny argument.
Scott Metzger’s The Bent Pinky for the 14th sees the return of anthropomorphic numerals humor. I’m a bit surprised Metzger goes so far as to make every numeral either a 3 or a 9. I’d have expected a couple of 2’s and 4’s. I understand not wanting to get into two-digit numbers. The premise of anthropomorphic numerals is troublesome if you need multiple-digit numbers.
Jon Rosenberg’s Goats for the 14th doesn’t directly mention a mathematical topic. But the story has the characters transported to a world with monkeys at typewriters. We know where that is. So we see that return after no time away, really.
Rick Detorie’s One Big Happy rerun for the 14th sees the return of “110 percent”. Happily the joke’s structured so that we can dodge arguing about whether it’s even possible to give 110 percent. I’m inclined to say of course it’s possible. “Giving 100 percent” in the context of playing a sport would mean giving the full reasonable effort. Or it does if we want to insist on idiomatic expressions making sense. It seems late to be insisting on that standard, but some people like it as an idea.
Terri Libenson’s Pajama Diaries for the 16th sees the return of Venn Diagram jokes. And it’s a properly-formed Venn Diagram, with the three circles coming together to indicate seven different conditions.
Gary Wise and Lance Aldrich’s Real Life Adventures for the 16th just name-drops rhomboids, using them as just a funny word. Geometry is filled with wonderful, funny-sounding words. I’m fond of “icosahedron” myself. But “rhomboid” and its related words are good ones. I think they hit that sweet spot between being uncommon in ordinary language without being so exotic that a reader’s eye trips over it. However funny a “triacontahedron” might be, no writer should expect the reader to forgive that pile of syllables. A rhomboid is a kind of parallelogram, so it’s got four sides. The sides come in two parallel pairs. Both members of a pair have the same length, but the different pairs don’t. They look like the kitchen tiles you’d get for a house you couldn’t really afford, not with tiling like that.
The last day of July and first day of August saw enough mathematically-themed comic strips to fill a standard-issue entry. The rest of the week wasn’t so well-stocked. But I’ll cover those comics on Tuesday if all goes well. This may be a silly plan, but it is a plan, and I will stick to that.
Johnny Hart’s Back To BC reprints the venerable and groundbreaking comic strip from its origins. On the 31st of July it reprinted a strip from February 1959 in which Peter discovers mathematics. The work’s elaborate, much more than we would use to solve the problem today. But it’s always like that. Newly-discovered mathematics is much like any new invention or innovation, a rickety set of things that just barely work. With time we learn better how the idea should be developed. And we become comfortable with the cultural assumptions going into the work. So we get more streamlined, faster, easier-to-use mathematics in time.
Mac King and Bill King’s Magic in a Minute for the 31st maybe isn’t really mathematics. I guess there’s something in the modular-arithmetic implied by it. But it depends on a neat coincidence. Follow the directions in the comic about picking a number from one to twelve and counting out the letters in the word for that number. And then the letters in the word for the number you’re pointing to, and then once again. It turns out this leads to the same number. I’d never seen this before and it’s neat that it does.
Rick Detorie’s One Big Happy rerun for the 31st features Ruthie teaching, as she will. She mentions offhand the “friendlier numbers”. By this she undoubtedly means the numbers that are attractive in some way, like being nice to draw. There are “friendly numbers”, though, as number theorists see things. These are sets of numbers. For each number in this set you get the same index if you add together all its divisors (including 1 and the original number) and divide it by the original number. For example, the divisors of six are 1, 2, 3, and 6. Add that together and you get 12; divide that by the original 6 and you get 2. The divisors of 28 are 1, 2, 4, 7, 14, and 28. Add that pile of numbers together and you get 56; divide that by the original 28 and you get 2. So 6 and 28 are friendly numbers, each the friend of the other.
As often happens with number theory there’s a lot of obvious things we don’t know. For example, we know that 1, 2, 3, 4, and 5 have no friends. But we do not know whether 10 has. Nor 14 nor 20. I do not know if it is proved whether there are infinitely many sets of friendly numbers. Nor do I know if it is proved whether there are infinitely many numbers without friends. Those last two sentences are about my ignorance, though, and don’t reflect what number theory people know. I’m open to hearing from people who know better.
There are also things called “amicable numbers”, which are easier to explain and to understand than “friendly numbers”. A pair of numbers are amicable if the sum of one number’s divisors is the other number. 220 and 284 are the smallest pair of amicable numbers. Fermat found that 17,296 and 18,416 were an amicable pair; Descartes found that 9,363,584 and 9,437,056 were. Both pairs were known to Arab mathematicians already. Amicable pairs are easy enough to produce. From the tenth century we’ve had Thâbit ibn Kurrah’s rule, which lets you generate sets of numbers. Ruthie wasn’t thinking of any of this, though, and was more thinking how much fun it is to write a 7.
Terry Border’s Bent Objects for the 1st just missed the anniversary of John Venn’s birthday and all the joke Venn Diagrams that were going around at least if your social media universe looks anything like mine.
Jon Rosenberg’s Scenes from a Multiverse for the 1st is set in “Mathpinion City”, in the “Numerically Flexible Zones”. And I appreciate it’s a joke about the politicization of science. But science and mathematics are human activities. They are culturally dependent. And especially at the dawn of a new field of study there will be long and bitter disputes about what basic terms should mean. It’s absurd for us to think that the question of whether 1 + 1 should equal 2 or 3 could even arise.
But we think that because we have absorbed ideas about what we mean by ‘1’, ‘2’, ‘3’, ‘plus’, and ‘equals’ that settle the question. There was, if I understand my mathematics history right — and I’m not happy with my reading on this — a period in which it was debated whether negative numbers should be considered as less than or greater than the positive numbers. Absurd? Thermodynamics allows for the existence of negative temperatures, and those represent extremely high-energy states, things that are hotter than positive temperatures. A thing may get hotter, from 1 Kelvin to 4 Kelvin to a million Kelvin to infinitely many Kelvin to -1000 Kelvin to -6 Kelvin. If there are intuition-defying things to consider about “negative six” then we should at least be open to the proposition that the universal truths of mathematics are understood by subjective processes.
When I split last week’s mathematically-themed comics I had just supposed there’d be some more on Friday to note. Live and learn, huh? Well, let me close out last week with a not-too-long essay. Better a couple of these than a few Reading the Comics posts long enough to break your foot on.
Adrian Raeside’s The Other Coastfor the 6th uses mathematics as a way to judge the fit and the unfit. (And Daryl isn’t even far wrong.) It’s understandable and the sort of thing people figure should flatter mathematicians. But it also plays on 19th-century social-Darwinist/eugenicist ideas which try binding together mental acuity and evolutionary “superiority”. It’s a cute joke but there is a nasty undercurrent.
Rich Powell’s Wide Open for the 7th shows a Western-style “Convolution Kid”. It’s shown here as just shouting numbers in-between a count so as to mess things up. That matches the ordinary definition and I’m amused with it as-is. Convolution is a good mathematical function, though one I don’t remember encountering until a couple years into my undergraduate career. It’s a binary operation, one that takes two functions and combines them into a new function. It turns out to be a natural way to understand signal processing. The original signal is one function. The way a processor changes a signal is another function. The convolution of the two is what actually comes out of the processing. Dividing this lets us study the behaviors of the processor separate from a particular problem.
And it turns up in other contexts. We can use convolution to solve differential equations, which turn up everywhere. We need to solve the differential equation for a special particular boundary condition, one called the Dirac delta function. That’s a really weird one. You have no idea. And it can require incredible ingenuity to find a solution. But once you have, you can find solutions for every boundary condition. You convolute the solution for the special case and the boundary condition you’re interested in, and there you go. The work may be particularly hard for this one case, but it is only the one case.
I knew my readership would drop off after I fell back from daily posting. Apparently it was worse than I imagined and nobody read my little blog here over the weekend. That’s fair enough; I had to tend other things myself. Still, for the purpose of maximizing the number of page views around here, taking two whole days off in a row was a mistake. There’s some more discussed in this Reading The Comics installment.
Word problems are dull. At least at the primary-school level. There’s all these questions about trains going in different directions or ropes sweeping out areas or water filling troughs. So Aaron McGruder’s Boondocks rerun from the 5th of May (originally run the 22nd of February, 2001) is a cute change. It’s at least the start of a legitimate word problem, based on the ways the recording industry took advantage of artists in the dismal days of fifteen years ago. I’m sure that’s all been fixed by now. Fill in some numbers and the question might interest people.
Glenn McCoy and Gary McCoy’s The Duplex for the 5th of May is a misunderstanding-fractions joke. I’m amused by the idea of messing up quarter-pound burgers. But it also brings to mind a summer when I worked for the Great Adventure amusement park and got assigned one day as cashier at the Great American Hamburger Stand. Thing is, I didn’t know anything about the stand besides the data point that they probably sold hamburgers. So customers would order stuff I didn’t know, and I couldn’t find how to enter it on the register, and all told it was a horrible mess. If you were stuck in that impossibly slow-moving line, I am sorry, but it was management’s fault; I told them I didn’t know what I was even selling. Also I didn’t know the drink cup sizes so I just charged you for whatever you said and if I gave you the wrong size I hope it was more soda than you needed.
On a less personal note, I have heard the claim about why one-third-pound burgers failed in United States fast-food places. Several chains tried them out in the past decade and they didn’t last, allegedly because too many customers thought a third of a pound was less than a quarter pound and weren’t going to pay more for less beef. It’s … plausible enough, I suppose, because people have never been good with fractions. But I suspect the problem is more linguistic. A quarter-pounder has a nice rhythm to it. A half-pound burger is a nice strong order to say. A third-pound burger? The words don’t even sound right. You have to say “third-of-a-pound burger” to make it seem like English, and it’s a terribly weak phrase. The fast food places should’ve put their money into naming it something that suggested big-ness but not too-big-to-eat.
Mark Tatulli’s Heart of the City for the 5th is about Heart’s dread of mathematics. Her expressed fear, that making one little mistake means the entire answer is wrong, is true enough. But how how much is that “enough”? If you add together someting that should be (say) 18, and you make it out to be 20 instead, that is an error. But that’s a different sort of error from adding them together and getting 56 instead.
And errors propagate. At least they do in real problems, in which you are calculating something because you want to use it for something else. An arithmetic error on one step might grow, possibly quite large, with further steps. That’s trouble. This is known as an “unstable” numerical calculation, in much the way a tin of picric acid dropped from a great height onto a fire is an “unstable” chemical. The error might stay about as large as it started out being, though. And that’s less troublesome. A mistake might stay predictable. The calculation is “stable” In a few blessed cases an error might be minimized by further calculations. You have to arrange the calculations cleverly to make that possible, though. That’s an extremely stable calculation.
And this is important because we always make errors. At least in any real calculation we do. When we want to turn, say, a formula like πr2 into a number we have to make a mistake. π is not 3.14, nor is it 3.141592, nor is it 3.14159265358979311599796346854418516. Does the error we make by turning π into some numerical approximation matter? It depends what we’re calculating, and how. There’s no escaping error and it might be a comfort to Heart, or any student, to know that much of mathematics is about understanding and managing error.
Joe Martin’s Boffo for the 6th of May is in its way about the wonder of very large numbers. On some reasonable assumptions — that our experience is typical, that nothing is causing traits to be concentrated one way or another — we can realize that we probably will not see any extreme condition. In this case, it’s about the most handsome men in the universe probably not even being in our galaxy. If the universe is large enough and people common enough in it, that’s probably right. But we likely haven’t got the least handsome either. Lacking reason to suppose otherwise we can guess that we’re in the vast middle.
David L Hoyt and Jeff Knurek’s Jumble for the 6th of May mentions mathematicians and that’s enough, isn’t it? Without spoiling the puzzle for anyone, I will say that “inocci” certainly ought to be a word meaning something. So get on that, word-makers.
Comic Strip Master Command had the regular pace of mathematically-themed comic strips the last few days. But it remembered what the 14th would be. You’ll see that when we get there.
Ray Billingsley’s Curtis for the 11th of March is a student-resists-the-word-problem joke. But it’s a more interesting word problem than usual. It’s your classic problem of two trains meeting, but rather than ask when they’ll meet it asks where. It’s just an extra little step once the time of meeting is made, but that’s all right by me. Anything to freshen the scenario up.
Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. for the 12th of March name-drops statisticians. Statisticians are almost expected to produce interesting pictures of their results. It is the field that gave us bar charts, pie charts, scatter plots, and many more. Statistics is, in part, about understanding a complicated set of data with a few numbers. It’s also about turning those numbers into recognizable pictures, all in the hope of finding meaning in a confusing world (ours).
Brian Anderson’s Dog Eat Doug for the 13th of March uses walls full of mathematical scrawl as signifier for “stuff thought deeply about’. I don’t recognize any of the symbols specifically, although some of them look plausibly like calculus. I would not be surprised if Anderson had copied equations from a book on string theory. I’d do it to tell this joke.
And then came the 14th of March. That gave us a bounty of Pi Day comics. Among them:
Edited To Add: And I forgot to mention, after noting to myself that I ought to mention it. The Price Is Right (the United States edition) hopped onto the Pi Day fuss. It used the day as a thematic link for its Showcase prize packages, noting how you could work out π from the circumference of your new bicycles, or how π was a letter from your vacation destination of Greece, and if you think there weren’t brand-new cars in both Showcases you don’t know the game show well. Did anyone learn anything mathematical from this? I am skeptical. Do people come away thinking mathematics is more fun after this? … Conceivably. At least it was a day fairly free of people declaring they Hate Math and Can Never Do It.
Bill Amend’s FoxTrot Classics for the 28th of November (originally run in 2004) depicts a “Christmas Card For Smart People”. It uses the familiar motif of “ability to do arithmetic” as denoting smartness. The key to the first word is remembering that mathematicians use the symbol ‘e’ to represent a number that’s just a little over 2.71828. We call the number ‘e’, or something ‘the base of the natural logarithm’. It turns up all over the place. If you have almost any quantity that grows or that shrinks at a speed proportional to how much there is, and describe how much of stuff there is over time, you’ll find an ‘e’. Leonhard Euler, who’s renowned for major advances in every field of mathematics, is also renowned for major advances in notation in physics, and he gave us ‘e’ for that number.
The key to the second word there is remembering from physics that force equals mass times acceleration. Therefore the force divided by the acceleration is …
And so that inspires this essay’s edition title. There are several comics in this selection that are about the symbols or the representations of mathematics, and that touch on the subject as a visual art.
Matt Janz’s Out of the Gene Pool for the 28th of November first ran the 26th of October, 2002. It would make for a good word problem, too, with a couple of levels: given the constraints of (a slightly looser) budget, how do they get the greatest number of cookies? Or if some cookies are better than others, how do they get the most enjoyment from their cookie purchase? Working out the greatest amount of enjoyment within a given cookie budget, with different qualities of cookies, can be a good introduction to optimization problems and how subtle they can be.
Bill Holbrook’s On The Fastrack for the 29th of November speaks in support of accounting. It’s a worthwhile message. It doesn’t get much respect, not from the general public, and not from typical mathematics department. The general public maybe thinks of accounting as not much more than a way companies nickel-and-dime them. If the mathematics departments I’ve associated with are fair representatives, accounting isn’t even thought of except by the assistant professor doing a seminar on financial mathematics. (And I’m not sure accounting gets mentioned there, since there’s exciting stuff about the Black-Scholes Equation and options markets to think about instead.) This despite that accounting is probably, by volume, the most used part of mathematics. Anyway, Holbrook’s strip probably won’t get the field a better reputation. But it has got some great illustrations of doing things with numbers. The folks in mathematics departments certainly have had days feeling like they’ve done each of these things.
Dave Coverly’s Speed Bump for the 30th of November is a compound interest joke. I admit I’ve told this sort of joke myself, proposing that the hour cut out of the day in spring when Daylight Saving Time starts comes back as a healthy hour and three minutes in autumn when it’s taken out of saving. If I can get the delivery right I might have someone going for that three minutes.
Mikael Wulff and Anders Morgenthaler’s Truth Facts for the 30th of November is a Venn diagram joke for breakfast. I would bet they’re kicking themselves for not making the intersection be the holes in the center.
Mark Anderson’s Andertoons for this week interests me. It uses a figure to try explaining how to relate gallon and quart an pint and other units relate to each other. I like it, but I’m embarrassed to say how long it took in my life to work out the relations between pints, quarts, gallons, and particularly whether the quart or the pint was the larger unit. I blame part of that on my never really having to mix a pint of something with a quart of something else, which ought to have sorted that out. Anyway, let’s always cherish good representations of information. Good representations organize information and relationships in ways that are easy to remember, or easy to reconstruct or extend.
John Graziano’s Ripley’s Believe It or Not for the 2nd of December tries to visualize how many ways there are to arrange a Rubik’s Cube. Counting off permutations of things by how many seconds it’d take to get through them all is a common game. The key to producing a staggering length of time is that it one billion seconds are nearly 32 years, and the number of combinations of things adds up really really fast. There’s over eight billion ways to draw seven letters in a row, after all, if every letter is equally likely and if you don’t limit yourself to real or even imaginable words. Rubik’s Cubes have a lot of potential arrangements. Graziano misspells Rubik, but I have to double-check and make sure I’ve got it right every time myself. I didn’t know that about the pigeons.
Charles Schulz’s Peanuts for the 2nd of December (originally run in 1968) has Peppermint Patty reflecting on the beauty of numbers. I don’t think it’s unusual to find some numbers particularly pleasant and others not. Some numbers are easy to work with; if I’m trying to add up a set of numbers and I have a 3, I look instinctively for a 7 because of how nice 10 is. If I’m trying to multiply numbers, I’d so like to multiply by a 5 or a 25 than by a 7 or an 18. Typically, people find they do better on addition and multiplication with lower numbers like two and three, and get shaky with sevens and eights and such. It may be quirky. My love is a wizard with 7’s, but can’t do a thing with 8. But it’s no more irrational than the way a person might a pyramid attractive but a sphere boring and a stellated icosahedron ugly.
I’ve seen some comments suggesting that Peppermint Patty is talking about numerals, that is, the way we represent numbers. That she might find the shape of the 2 gentle, while 5 looks hostile. (I can imagine turning a 5 into a drawing of a shouting person with a few pencil strokes.) But she doesn’t seem to say one way or another. She might see a page of numbers as visual art; she might see them as wonderful things with which to play.
Some of the past several days’ mathematically-themed comic strips have bits of wordplay in them. That’ll do for the theme. We get some familiar topics along the way.
Rick Detorie’s One Big Happy for the 6th of October is one of the wordplay jokes you can do about probability. (This is the strip that ran in newspapers this year. One Big Happy strips on Gocomics.com are reruns from several years back.)
Tom Thaves’s Frank and Ernest for the 8th of October is a kids-resisting-algebra problem. The kid asks why ‘x’ has to be equal to something, why it can’t just be ‘x’. He’s wiser than his teacher has taught. We use ‘x’ as the name for a number whose exact identity we don’t know right away. Often, especially in introductory algebra, we hope to work out what number it is. That’s the sort of problem that makes us find x, or solve for x. But we don’t always care what x is. Sometimes we just want to say that it’s an example of a number with some interesting properties. We often use it this way when we try drawing the plot of a function. The plot shows all the coordinate sets that make some equation true, and we need x to organize our thoughts about that, but we never really care what x is.
Or we might use x as a ‘dummy variable’, the mathematical equivalent of falsework. We use the variable to get some work done, but never see it once we’re finished, and don’t ever care what it was. If we take the definite integral of a function of x over x, for example, the one thing our answer should not have is an ‘x’ in it. (Well, if we’re integrating some nasty function that can’t be evaluated except in terms of another integral maybe an ‘x’ will appear. But that’s a pathological case.)
Alternatively, x might be a parameter, something which has to be a fixed number for the sake of doing other work, but whose value we don’t really care about. This would be an eccentric choice — usually parameters are from earlier in the alphabet, rarely later than ‘l’ and almost never past ‘t’ — but sometimes that’s the best alternative.
In Jef Mallett’s Frazz for the 8th of October, Caulfield answers his teacher’s demand to “show his work” by presenting a slide rule. It’s a cute joke although I’m not on Caulfield’s side here. If all anyone cared about was whether the calculation was right we’d need no mathematics. We have computers. What is worth teaching is “how do you know what to compute”, with a sideline of “can you do the computations correctly”. It’s important to know what you mean to do. It’s also important to know how to plausibly find an answer if you don’t know exactly what to do. None of that is shown by the answer alone.
Disney’s Donald Duck for the 10th of October, a rerun from goodness knows when, depicts accurately the most terrifying moment a mathematician endures. I am delighted to see that the equations written out are correct and even consistent from one panel to the next. And yes, real mathematicians will sometimes write down what seem like altogether too-obvious propositions. That’s a good way of making sure you aren’t tripping over the easy stuff on the way to the bigger conclusions. I think it’s a bit implausible that the entire board would be this level of stuff — by the time you have your PhD, at least in mathematics or physics, you don’t need help remembering what the cosine of 120 degrees is — but it’s all valid stuff. Well, I could probably use the help remembering the tangent angle-addition formula, if I ever needed to work out the tangent of the sum of two angles.
The good news is I’ve got a couple of comic strips I feel responsible including the pictures for. (While I’m confident I could include all the comics I talk about as fair use — I make comments which expand on the strips’ content and which don’t make sense without the original — Gocomics.com links seem reasonably stable and likely to be there in the future. Comics Kingdom links generally expire after a month except to subscribers and I don’t know how long Creators.com links last.) And a couple of them talk about rock bands, so, that’s why I picked that titel.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 17th of September is a subverted-fairy-tale-moral strip, naturally enough. It’s also a legitimate point, though. Unlikely events do happen sometimes, and it’s a mistake to draw too-strong conclusions from them. This is why it’s important to reproduce interesting results. It’s also why, generally, we like larger sample sizes. It’s not likely that twenty fair coins flipped will all come up tails at once. But it’s far more likely that will happen than that two hundred fair coins flipped will all come up tails. And that’s far more likely than that two thousand fair coins will. For that matter, it’s more likely that three-quarters of twenty fair coins flipped will come up tails than that three-quarters of two hundred fair coins will. And the chance that three-quarters of two thousand fair coins will come up tails is ignorable. If that happens, then something interesting has been found.
In Juba’s Viivi and Wagner for the 17th of September, Wagner announces his decision to be a wandering mathematician. I applaud his ambition. If I had any idea where to find someone who needed mathematics done I’d be doing that myself. If you hear something give me a call. I’ll be down at the City Market, in front of my love’s guitar case, multiplying things by seven. I may get it wrong, but nobody will know how to correct me.
Daniel Beyer’s Long Story Short for the 18th of September uses a page full of calculations to predict when prog-rock band Tool will release their next album. (Wikipedia indicates they’re hoping for sometime before the end of 2015, but they’ve been working on it since 2008.) Some of the symbols make a bit of sense as resembling those of quantum physics. An expression like (in the lower left of the board) resembles a probability distribution calculation. (There should be a ^ above the H there, but that’s a little beyond what WordPress can render in the simple mathematical LaTeX tools it has available. It’s in the panel, though.) The letter ψ stands for a probability wave, containing somehow all the information about a system. The composition of symbols literally means to calculate how an operator — a function that has a domain of functions and a range of functions — changes that probability distribution. In quantum mechanics every interesting physical property has a matching operator, and calculating this set of symbols tells us the distribution of whatever that property is. H generally suggests the total energy of the system, so the implication is this measures, somehow, what energies are more and are less probable. I’d be interested to know if Beyer took the symbols from a textbook or paper and what the original context was.
Dave Whamond’s Reality Check for the 19th of September brings in another band to this review. It uses a more basic level of mathematics, though.
Percy Crosby’s Skippy from the 19th of September — rerun from sometime in 1928 — is a clever way to get a word problem calculated. It also shows off what’s probably been the most important use of arithmetic, which is keeping track of money. Accountants and shopkeepers get little attention in histories of mathematics, but a lot of what we do has been shaped by their needs for speed, efficiency, and accuracy. And one of Gocomics’s commenters pointed out that the shopkeeper didn’t give the right answer. Possibly the shopkeeper suspected what was up.
Paul Trap’s Thatababy for the 20th of September uses a basic geometry fact as an example of being “very educated”. I don’t think the area of the circle rises to the level of “very” — the word means “truly”, after all — but I would include it as part of the general all-around awareness of the world people should have. Also it fits in the truly confined space available. I like the dad’s eyes in the concluding panel. Also, there’s people who put eggplant on pizza? Really? Also, bacon? Really?
Alex Hallatt’s Arctic Circle for the 21st of September is about making your own luck. I find it interesting in that it rationalizes magic as a thing which manipulates probability. As ways to explain magic for stories go that isn’t a bad one. We can at least imagine the rigging of card decks and weighting of dice. And its plot happens in the real world, too: people faking things — deceptive experimental results, rigged gambling devices, financial fraud — can often be found because the available results are too improbable. For example, a property called Benford’s Law tells us that in many kinds of data the first digit is more likely to be a 1 than a 2, a 2 than a 3, a 3 than a 4, et cetera. This fact serves to uncover fraud surprisingly often: people will try to steal money close to but not at some limit, like the $10,000 (United States) limit before money transactions get reported to the federal government. But that means they work with checks worth nine thousand and something dollars much more often than they do checks worth one thousand and something dollars, which is suspicious. Randomness can be a tool for honesty.
Peter Maresca’s Origins of the Sunday Comics feature for the 21st of September ran a Rube Goldberg comic strip from the 19th of November, 1913. That strip, Mike and Ike, precedes its surprisingly grim storyline with a kids-resisting-the-word-problem joke. The joke interests me because it shows a century-old example of the joke about word problems being strings of non sequiturs stuffed with unpleasant numbers. I enjoyed Mike and Ike’s answer, and the subversion of even that answer.
Mark Anderson’s Andertoons for the 22nd of September tries to optimize its targeting toward me by being an anthropomorphized-mathematical-objects joke and a Venn diagram joke. Also being Mark Anderson’s Andertoons today. If I didn’t identify this as my favorite strip of this set Anderson would just come back with this, but featuring monkeys at typewriters too.