Last week saw a modest number of mathematically-themed comic strips. Then it threw in a bunch of them all on Thursday. I’m splitting the week partway through that, since it gives me some theme to this collection.
Tim Rickard’s Brewster Rockit for the 3rd of May is a dictionary joke, with Brewster naming each kind of chart and making a quick joke about it. The comic may help people who’ve had trouble remembering the names of different kinds of graphs. I doubt people are likely to confuse a pie chart with a bar chart, admittedly. But I could imagine thinking a ‘line graph’ is what we call a bar chart, especially if the bars are laid out horizontally as in the second panel here.
The point of all these graphs is to understand data geometrically. We have fair intuitions about relatives lengths and areas. Bar charts represent relative magnitudes in lengths. Pie charts and bubble charts represent magnitudes in area. We have okay skills in noticing structures in complex shapes. Line graphs and scatter plots use that skill. So these pictures can help us understand some abstraction or something we can’t sense using a sense we do have. It’s not necessarily great; note that I said our intuitions were ‘fair’ and ‘okay’. But we hope to use reason helped by intuition to better understand what we are doing.
And, yes, in the greater scheme of things, any homework or classwork problem is trivial. It’s meant to teach how to calculate things we would like to know. The framing of the story is meant to give us a reason to want to know a thing. But they are practice, and meant to be practice. One practices on something of no consequence, where errors in one’s technique can be corrected without breaking anything.
It happens a round of story problems broke out among my family. My sister’s house has some very large trees. There turns out to be a poorly-organized process for estimating the age of these trees from their circumference. This past week saw a lot of chatter and disagreement about what the ages of these trees might be.
Michael Fry’s Committed rerun for the 7th finally gets us to golf. The Lazy Parent tries to pass off watching golf as educational, with working out the distance to the pin as a story problem. Structurally this is just fine, though: a golfer would be interested to know how far the ball has yet to go. All the information needed is given. It’s the question of whether anyone but syndicated cartoonists cares about golf that’s a mystery.
I’m again falling behind the comic strips; I haven’t had the writing time I’d like, and that review of last month’s readership has to go somewhere. So let me try to dig my way back to current. The happy news is I get to do one of those single-day Reading the Comics posts, nearly.
Harley Schwadron’s 9 to 5 for the 7th strongly implies that the kid wearing a lemon juicer for his hat has nearly flunked arithmetic. At the least it’s mathematics symbols used to establish this is a school.
Jef Mallett’s Frazz for the 7th has kids thinking about numbers whose (English) names rhyme. And that there are surprisingly few of them, considering that at least the smaller whole numbers are some of the most commonly used words in the language. It would be interesting if there’s some deeper reason that they don’t happen to rhyme, but I would expect that it’s just, well, why should the names of 6 and 8 (say) have anything to do with each other?
There are, arguably, gaps in Evan and Kevyn’s reasoning, and on the 8th one of the other kids brings them up. Basically, is there any reason to say that thirteen and nineteen don’t rhyme? Or that twenty-one and forty-one don’t? Evan writes this off as pedantry. But I, admittedly inclined to be a pedant, think there’s a fair question here. How many numbers do we have names for? Is there something different between the name we have for 11 and the name we have for 1100? Or 2011?
There isn’t an objectively right or wrong answer; at most there are answers that are more or less logically consistent, or that are more or less convenient. Finding what those differences are can be interesting, and I think it bad faith to shut down the argument as “pedantry”.
Dave Whamond’s Reality Check for the 7th claims “birds aren’t partial to fractions” and shows a bird working out, partially with diagrams, the saying about birds in the hand and what they’re worth in the bush.
The narration box, phrasing the bird as not being “partial to fractions”, intrigues me. I don’t know if the choice is coincidental on Whamond’s part. But there is something called “partial fractions” that you get to learn painfully well in Calculus II. It’s used in integrating functions. It turns out that you often can turn a “rational function”, one whose rule is one polynomial divided by another, into the sum of simpler fractions. The point of that is making the fractions into things easier to integrate. The technique is clever, but it’s hard to learn. And, I must admit, I’m not sure I’ve ever used it to solve a problem of interest to me. But it’s very testable stuff.
There were a handful of other comic strips last week. If they have a common theme (and I’ll try to drag one out) it’s that they circle around pragmatism. Not just using mathematics in the real world but the fussy stuff of what you can calculate and what you can use a calculation for.
And, again, I am hosting the Playful Math Education Blog Carnival this month. If you’ve run across any online tool that teaches mathematics, or highlights some delightful feature of mathematics? Please, let me know about it here, and let me know what of your own projects I should feature with it. The goal is to share things about mathematics that helped you understand more of it. Even if you think it’s a slight thing (“who cares if you can tell whether a number’s divisible by 11 by counting the digits right?”) don’t worry. Slight things count. Speaking of which …
Jef Mallett’s Frazz for the 20th has a kid ask about one of those add-the-digits divisibility tests. What happens if the number is too big to add up all the digits? In some sense, the question is meaningless. We can imagine finding the sum of digits no matter how many digits there are. At least if there are finitely many digits.
But there is a serious mathematical question here. We accept the existence of numbers so big no human being could ever know their precise value. At least, we accept they exist in the same way that “4” exists. If a computation can’t actually be finished, then, does it actually mean anything? And if we can’t figure a way to shorten the calculation, the way we can usually turn the infinitely-long sum of a series into a neat little formula?
This gets into some cutting-edge mathematics. For calculations, some. But also, importantly, for proofs. A proof is, really, a convincing argument that something is true. The ideal of this is a completely filled-out string of logical deductions. These will take a long while. But, as long as it takes finitely many steps to complete, we normally accept the proof as done. We can imagine proofs that take more steps to complete than could possibly be thought out, or checked, or confirmed. We, living in the days after Gödel, are aware of the idea that there are statements which are true but unprovable. This is not that. Gödel’s Incompleteness Theorems tell us about statements that a deductive system can’t address. This is different. This is things that could be proven true (or false), if only the universe were more vast than it is.
There are logicians who work on the problem of what too-long-for-the-universe proofs can mean. Or even what infinitely long proofs can mean, if we allow those. And how they challenge our ideas of what “proof” and “knowledge” and “truth” are. I am not among these people, though, and can’t tell you what interesting results they have concluded. I just want to let you know the kid in Frazz is asking a question you can get a spot in a mathematics or philosophy department pondering. I mean so far as it’s possible to get a spot in a mathematics or philosophy department.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th is a less heady topic. Its speaker is doing an ethical calculation. These sorts of things are easy to spin into awful conclusions. They treat things like suffering with the same tools that we use to address the rates of fluids mixing, or of video game statistics. This often seems to trivialize suffering, which we feel like we shouldn’t do.
This kind of calculation is often done, though. It’s rather a hallmark of utilitarianism to try writing an equation for an ethical question. It blends often more into economics, where the questions can seem less cruel even if they are still about questions of life and death. But as with any model, what you build into the model directs your results. The lecturer here supposes that guilt is diminished by involving more people. (This seems rather true to human psychology, though it’s likely more that the sense of individual responsibility dissolves in a large enough group. There are many other things at work, though, all complicated and interacting in nonlinear ways.) If we supposed that the important measure was responsibility for the killing, we would get that the more people involved in killing, the worse it is, and that a larger war only gets less and less ethical. (This also seems true to human psychology.)
Jeff Corriveau’s Deflocked for the 20th sees Mamet calculating how many days of life he expects to have left. There are roughly 1,100 days in three years, so, Mamet’s figuring on about 40 years of life. These kinds of calculation are often grim to consider. But we all have long-term plans that we would like to do (retirement, and its needed savings, are an important one) and there’s no making a meaningful plan without an idea of what the goals are.
This finally closes out the last week’s comic strips. Please stop in next week as I get to some more mathematics comics and the Playful Math Education Blog Carnival. Thanks for reading.
You know, I had picked these comic strips out as the ones that, last week, had the most substantial mathematics content. And on preparing this essay I realize there’s still not much. Maybe I could have skipped out on the whole week instead.
Bill Amend’s FoxTrot for the 1st is mostly some wordplay. Jason’s finding ways to represent the counting numbers with square roots. The joke plays more tightly than one might expect. Root beer was, traditionally, made with sassafras root, hence the name. (Most commercial root beers don’t use actual sassafras anymore as the safrole in it is carcinogenic.) The mathematical term root, meanwhile, derives from the idea that the root of a number is the thing which generates it. That 2 is the fourth root of 16, because four 2’s multiplied together is 16. That idea. This draws on the metaphor of the roots of a plant being the thing which lets the plant grow. This isn’t one of those cases where two words have fused together into one set of letters.
Jef Mallett’s Frazz for the 1st is set up with an exponential growth premise. The kid — I can’t figure out his name — promises to increase the number of push-ups he does each day by ten percent, with exciting forecasts for how many that will be before long. As Frazz observes, it’s not especially realistic. It’s hard to figure someone working themselves up from nothing to 300 push-ups a day in only two months.
Also much else of the kid’s plan doesn’t make sense. On the second day he plans to do 1.1 push-ups? On the third 1.21 push-ups? I suppose we can rationalize that, anyway, by taking about getting a fraction of the way through a push-up. But if we do that, then, I make out by the end of the month that he’d be doing about 15.863 push-ups a day. At the end of two months, at this rate, he’d be at 276.8 push-ups a day. That’s close enough to three hundred that I’d let him round it off. But nobody could be generous enough to round 15.8 up to 90.
An alternate interpretation of his plans would be to say that each day he’s doing ten percent more, and round that up. So that, like, on the second day he’d do 1.1 rounded up to 2 push-ups, and on the third day 2.2 rounded up to 3 push-ups, and so on. Then day thirty looks good: he’d be doing 94. But the end of two months is a mess as by then he’d be doing 1,714 push-ups a day. I don’t see a way to fit all these pieces together. I’m curious what the kid thought his calculation was. Or, possibly, what Jef Mallett thought the calculation was.
Zach Weinersmith’s for the 2nd has a kid rejecting accounting in favor of his art. But, wanting to do that art with optimum efficiency … ends up doing accounting. It’s a common story. A common question after working out that someone can do a thing is how to do it best. Best has many measures, yes. But the logic behind how to find it stays the same. Here I admit my favorite kinds of games tend to have screen after screen of numbers, with the goal being to make some number as great as possible considering. If they ever made Multiple Entry Accounting Simulator none of you would ever hear from me again.
Which may be some time! Between Reading the Comics, A to Z, recap posts, and the occasional bit of filler I’ve just finished slightly over a hundred days in a row posting something. That is, however, at its end. I don’t figure to post anything tomorrow. I may not have anything before Sunday’s Reading the Comics post, at this link. I’ll be letting my typing fingers sleep in instead. Thanks for reading.
There were more mathematically-themed comic strips last week than I had time to deal with. This is in part because of something Saturday which took several more hours than I had expected. So let me start this week with some of the comics that, last week, mentioned mathematics in a marginal enough way there’s nothing to say about them besides yeah, that’s a comic strip which mentioned mathematics.
Jef Mallett’s Frazz for the 27th has a kid wondering why they need in-person instruction for arithmetic. (I’d agree that rehearsing arithmetic skills is very easy to automate. You can make practice problems pretty near without limit. How much this has to do with mathematics is a point of debate.)
This is almost all a post about some comics that don’t need more than a mention. You know, strips that just have someone in class not buying the word problem. These are the rest of last week’s.
Before I get there, though, I want to share something. I ran across an essay by Chris K Caldwell and Yeng Xiong: What Is The Smallest Prime? The topic is about 1, and whether that should be a prime number. Everyone who knows a little about mathematics knows that 1 is generally not considered a prime number. But we’re also a bit stumped to figure out why, since the idea of “a prime number is divisible by 1 and itself” seems to fit this, even if the fit is weird. And we have an explanation for this: 1 used to be thought of as prime, but it made various theorems more clumsy to present. So it was either cut 1 out of the definition or add the equivalent work to everything, and mathematicians went for the solution that was less work. I know that I’ve shared this story around here. (I’m surprised to find I didn’t share it in my Summer 2017 A-to-Z essay about prime numbers.)
The truth is more complicated than that. The truth of anything is always more complicated than its history. Even an excellent history’s. It’s not that the short story has things wrong, precisely. But that that matters are more complicated than that. The history includes things we forget were ever problems, like, the question of whether 1 should be a number. And that the question of whether mathematicians “used to” consider 1 a number is built on the supposition that mathematicians were a lot more uniform in their thinking than they were. Even to the individual: people were inconsistent in what they themselves wrote, because most mathematicians turn out to be people.
Tim Rickard’s Brewster Rockit for the 17th mentions entropy, which is so central to understanding statistical mechanics and information theory. It’s in the popular understanding of entropy, that of it being a thing which makes stuff get worse. But that’s of mathematical importance too.
The first, important, thing is that I have not disappeared or done something worse. I just had one of those weeks where enough was happening that something had to give. I could either write up stuff for my mathematics blog, or I could feel guilty about not writing stuff up for my mathematics blog. Since I didn’t have time to do both, I went with feeling guilty about not writing, instead. I’m hoping this week will give me more writing time, but I am fooling only myself.
Second is that Comics Kingdom has, for all my complaining, gotten less bad in the redesign. Mostly in that the whole comics page loads at once, now, instead of needing me to click to “load more comics” every six strips. Good. The strips still appear in weird random orders, especially strips like Prince Valiant that only run on Sundays, but still. I can take seeing a vintage Boner’s Ark Sunday strip six unnecessary times. The strips are still smaller than they used to be, and they’re not using the decent, three-row format that they used to. And the archives don’t let you look at a week’s worth in one page. But it’s less bad, and isn’t that all we can ever hope for out of the Internet anymore?
And finally, Comic Strip Master Command wanted to make this an easy week for me by not having a lot to write about. It got so light I’ve maybe overcompensated. I’m not sure I have enough to write about here, but, I don’t want to completely vanish either.
Dave Whamond’s Reality Check for the 15th is … hm. Well, it’s not an anthropomorphic-numerals joke. It is some kind of wordplay, making concrete a common phrase about, and attitude toward, numbers. I could make the fussy difference between numbers and numerals here but I’m not sure anyone has the patience for that.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 17th touches around mathematics without, I admit, necessarily saying anything specific. The angel(?) welcoming the man to heaven mentions creating new systems of mathematics as some fit job for the heavenly host. The discussion of creating self-consistent physics systems seems mathematical in nature too. I’m not sure whether saying one could “attempt” to create self-consistent physics is meant to imply that our universe’s physics are not self-consistent. To create a “maximally complex reality using the simplest possible constructions” seems like a mathematical challenge as well. There are important fields of mathematics built on optimizing, trying to create the most extreme of one thing subject to some constraints or other.
I think the strip’s premise is the old, partially a joke, concept that God is a mathematician. This would explain why the angel(?) seems to rate doing mathematics or mathematics-related projects as so important. But even then … well, consider. There’s nothing about designing new systems of mathematics that ordinary mortals can’t do. Creating new physics or new realities is beyond us, certainly, but designing the rules for such seems possible. I think I understood this comic better then I had thought about it less. Maybe including it in this column has only made trouble for me.
Doug Savage’s Savage Chickens for the 17th amuses me by making a strip out of a logic paradox. It’s not quite your “this statement is a lie” paradox, but it feels close to that, to me. To have the first chicken call it “Birthday Paradox” also teases a familiar probability problem. It’s not a true paradox. It merely surprises people who haven’t encountered the problem before. This would be the question of how many people you need to have in a group before there’s a 50 percent (75 percent, 99 percent, whatever you like) chance of at least one pair sharing a birthday.
And I notice on Wikipedia a neat variation of this birthday problem. This generalization considers splitting people into two distinct groups, and how many people you need in each group to have a set chance of a pair, one person from each group, sharing a birthday. Apparently both a 32-person group of 16 women and 16 men, or a 49-person group of 43 women and six men, have a 50% chance of some woman-man pair sharing a birthday. Neat.
Mark Parisi’s Off The Mark for the 18th sports a bit of wordplay. It’s built on how multiplication and division also have meanings in biology. … If I’m not mis-reading my dictionary, “multiply” meant any increase in number first, and the arithmetic operation we now call multiplication afterwards. Division, similarly, meant to separate into parts before it meant the mathematical operation as well. So it might be fairer to say that multiplication and division are words that picked up mathematical meaning.
It’s happened again: another slow week around here. My supposition is that Comic Strip Master Command was snowed in about a month ago, and I’m seeing the effects only now. There’s obviously no other reason that more comic strips didn’t address my particular narrow interest in one seven-day span.
Samson’s Dark Side of the Horse for the 18th is a numerals joke. The mathematics content is slight, I admit, but I’ve always had a fondness for Dark Side of the Horse. (I know it sounds like I have a fondness for every comic strip out there. I don’t quite, but I grant it’s close.) Conflating numerals and letters, and finding words represented by numerals, is an old tradition. It was more compelling in ancient days when letters were used as numerals so that it was impossible not to find neat coincidences. I suppose these days it’s largely confined to typefaces that make it easy to conflate a letter and a numeral. I mean moreso than the usual trouble telling apart 1 and l, 0 and O, or 5 and S. Or to special cases like hexadecimal numbers where, for ease of representation, we use the letters A through F as numerals.
Jef Mallett’s Frazz for the 18th is built on an ancient problem. I remember being frustrated with it. How is “questions 15 to 25” eleven questions when the difference between 15 and 25 is ten? The problem creeps into many fields. Most of the passion has gone out of the argument but around 1999 you could get a good fight going about whether the new millennium was to begin with January 2000 or 2001. The kind of problem is called a ‘fencepost error’. The name implies how often this has complicated someone’s work. Divide a line into ten segments. There are nine cuts on the interior of the line and the two original edges. I’m not sure I could explain to an elementary school student how the cuts and edges of a ten-unit-long strip match up to the questions in this assignment. I might ask how many birthdays someone’s had when they’re nine years old, though. And then flee the encounter.
Mark Parisi’s Off The Mark for the 19th is another numerals joke. This one’s also the major joke to make about an ice skater doing a figure eight: write the eight some other way. (I’d have sworn there was an M-G-M Droopy cartoon in which Spike demonstrates his ability to skate a figure 8, and then Droopy upstages him by skating ‘4 + 4’. I seem to be imagining it; the only cartoon where this seems to possibly fit is 1950’s The Chump Champ, and the joke isn’t in that one. If someone knows the cartoon I am thinking of, please let me know.) Here, the robot is supposed to be skating some binary numeral. It’s nothing close to an ‘8’, but perhaps the robot figures it needs to demonstrate some impressive number to stand out.
Bud Blake’s Tiger for the 21st has Tiger trying to teach his brother arithmetic. Working it out with fingers seems like a decent path to try, given Punkinhead’s age and background. And Punkinhead has a good point: why is the demonstration the easy problem and the homework the hard problem? I haven’t taught in a while, but do know I would do that sort of thing. My rationalization, I think, would be that a hard problem is usually hard because it involves several things. If I want to teach a thing, then I want to highlight just that thing. So I would focus on a problem in which that thing is the only tricky part, and everything else is something the students are so familiar with they don’t notice it. The result is usually an easy problem. There isn’t room for toughness. I’m not sure if that’s a thing I should change, though. Demonstrations of how to work harder problems are worth doing. But I usually think of those as teaching “how to use these several things we already know”. Using a tough problem to show one new thing, plus several already-existing tricky things, seems dangerous. It might be worth it, though.
I had not wanted to mention, for fear of setting off a panic. But Mark Anderson’s Andertoons, which I think of as being in every Reading the Comics post, hasn’t been around lately. If I’m not missing something, it hasn’t made an appearance in three months now. I don’t know why, and I’ve been trying not to look too worried by it. Mostly I’ve been forgetting to mention the strange absence. This even though I would think any given Tuesday or Friday that I should talk about the strip not having anything for me to write about. Fretting about it would make a great running theme. But I have never spotted a running theme before it’s finished. In any event the good news is that the long drought has ended, and Andertoons reappears this week. Yes, I’m hoping that it won’t be going to long between appearances this time.
Jef Mallett’s Frazz for the 16th talks about probabilities. This in the context of assessing risks. People are really bad at estimating probabilities. We’re notoriously worse at assessing risks, especially when it’s a matter of balancing a present cost like “fifteen minutes waiting while the pharmacy figures out whether insurance will pay for the flu shot” versus a nebulous benefit like “lessened chance of getting influenza, or at least having a less severe influenza”. And it’s asymmetric, too. We view improbable but potentially enormous losses differently from the way we view improbable but potentially enormous gains. And it’s hard to make the rationally-correct choice reliably, not when there are so many choices of this kind every day.
Tak Bui’s PC and Pixel for the 16th features a wall full of mathematical symbols, used to represent deep thought about a topic. The symbols are gibberish, yes. I’m not sure that an actual “escape probability” could be done in a legible way, though. Or even what precisely Professor Phillip might be calculating. I imagine it would be an estimate of the various ways he might try to escape, and what things might affect that. This might be for the purpose of figuring out what he might do to maximize his chances of a successful escape. Although I wouldn’t put it past the professor to just be quite curious what the odds are. There’s a thrill in having a problem solved, even if you don’t use the answer for anything.
Ruben Bolling’s Super-Fun-Pak Comix for the 18th has a trivia-panel-spoof dubbed Amazing Yet Tautological. One could make an argument that most mathematics trivia fits into this category. At least anything about something that’s been proven. Anyway, whether this is a tautological strip depends on what the strip means by “average” in the phrase “average serving”. There’s about four jillion things dubbed “average” and each of them has a context in which they make sense. The thing intended here, and the thing meant if nobody says anything otherwise, is the “arithmetic mean”. That’s what you get from adding up everything in a sample (here, the amount of egg salad each person in America eats per year) and dividing it by the size of the sample (the number of people in America that year). Another “average” which would make sense, but would break this strip, would be the median. That would be the amount of egg salad that half of all Americans eat more than, and half eat less than. But whether every American could have that big a serving really depends on what that median is. The “mode”, the most common serving, would also be a reasonable “average” to expect someone to talk about.
Mark Anderson’s Andertoons for the 19th is that strip’s much-awaited return to my column here. It features solid geometry, which is both an important part of geometry and also a part that doesn’t get nearly as much attention as plane geometry. It’s reductive to suppose the problem is that it’s harder to draw solids than planar figures. I suspect that’s a fair part of the problem, though. Mathematicians don’t get much art training, not anymore. And while geometry is supposed to be able to rely on pure reasoning, a good picture still helps. And a bad picture will lead us into trouble.
It’s another week with several on-topic installments of Frazz. Again, Jef Mallet, you and I live in the same metro area. Wave to me at the farmer’s market or something. I’m kind of able to talk to people in real life, if I can keep in view three different paths to escape and know two bathrooms to hide in. Horrock’s is great for that.
Jef Mallet’s Frazz for the 22nd is a bit of wordplay. It’s built on the association between “negative” and “wrong”. And the confusing fact that multiplying a negative number by a negative number results in a positive number. It sounds like a trick. Still, negative numbers are tricky. The name connotes something that’s gone a bit wrong. It took time to understand what they were and how they should work. This weird multiplication rule follows from that. If we don’t suppose this to be true, then we break other ideas we have about multiplication and comparative sizes and such. Mathematicians needed to get comfortable with negative numbers. For a long time, for example, mathematicians would treat and as different kinds of polynomials to solve. Today we see a -4 as no harder than a +4, now that we’re good at multiplying it out. And I have read, but have not seen explained, that there was uncertainty among the philosophers of mathematics about whether we should consider negative numbers, as a group, to be greater than or less than positive numbers. (I have reasons for thinking this a mighty interesting speculation.) There’s reasons to doubt them, is what I have to say.
Bob Weber Jr and Jay Stephens’s Oh Brother for the 22nd reminds me of my childhood. At some point I was pairing up the counting numbers and the letters of the alphabet, and realized that the alphabet ended while the numbers did not. Something about that offended my young sense of justice. I’m not sure how, anymore. But that it was always possible to find a bigger number than whatever you thought was the biggest caught my imagination.
There is, surely, a largest finite number that anybody will ever use for something, even if it’s just hyperbole. I’m curious what it will be. Surely we can’t have already used it. A number named Skewes’s Number was famous, for a while, as the largest number actually used in a proof of something. The fame came from Isaac Asimov writing an essay about the number, and why someone might care, and how hard it is just describing how big the number is in a comprehensible way. Wikipedia tells me this number’s far been exceeded by, among other things, something called Rayo’s Number. It’s “the smallest number bigger than any finite number named by an expression in the language of set theory with a googol symbols or less” (plus some technical points to keep you from cheating). Which, all right, but I’d like to know if we think the first digit is a 1, maybe a 2? Somehow I don’t demand that of Skewes, perhaps because I read that Asimov essay when I was at an impressionable age.
Jef Mallet’s Frazz for the 23rd has Caulfield talk about a fraction divided by a fraction. And particularly he says “a fraction divided by a fraction is just a fraction times a flipped fraction”. This offends me, somehow. This even though that is how I’d calculate the value of the division, if I needed to know that. But it seems to me like automatically going to that process skips recognizing that, say, shouldn’t be surprising if it turns out not to be a fraction. Well, Caulfield’s just looking to cause trouble with a string of wordplay. I can think of how to divide a fraction by a fraction and get zero.
Ashleigh Brilliant’s Pot-Shots for the 23rd promises to recapitulate the whole history of mathematics in a single panel. Ambitious bit of work. It’s easy to picture going from the idea of 1 to any of the positive whole numbers, though. It’s so easy it doesn’t even need humans to do it; animals can count, at least a bit. We just carry on to a greater extent than the crows or the raccoons do, so far as we’ve heard. From those, it takes some squinting, but you can think of negative whole numbers. And from that you get zero pretty quickly. You can also get rational numbers. The western mathematical tradition did this by looking at … er … ratios, that something might be to another thing as two is to five. Circumlocutions like that. Getting to irrational numbers is harder. Can be harder. Some irrational numbers beg you to notice them: the square root of two, for example. Square root of three. Numbers that come up from solving polynomial equations. But there are more number than those. Many more numbers. You might suspect the existence of a transcendental number, that isn’t the root of any polynomial that’s decently behaved. But finding one? Or finding that there are more transcendental number than there are real numbers? This takes a certain brilliance to suspect, and to prove out. But we can get there with rational numbers — which we get to from collections of ones — and the idea of cutting sets of numbers into those smaller than and those bigger than something. Ashleigh Brilliant has more truth than, perhaps, he realized when he drew this panel.
Niklas Eriksson’s Carpe Diem for the 24th has goldfish work out the shape of space. A goldfish in this case has the advantage of being able to go nearly everywhere in the space. But working out what the universe must look like, when you can only run local experiments, is a great geometric problem. It’s akin to working out that the Earth must be a sphere, and about how big a sphere, from the surveying job one can do without travelling more than a few hundred kilometers.
Sandra Bell-Lundy’s Between Friends for the 1st is a Venn Diagram joke to start off the week. The form looks wrong, though. This can fool the reader into thinking the cartoonist messed up the illustration. Here’s why. The point of a Venn Diagram is to show the two or more groups of things and identify what they have in common. It is true that any life will have regrets about things done. And regrets about things not done. But what are the things that one both ‘did do’ and ‘didn’t do’? Unless you accept the weasel-wording of “did halfheartedly”, there is nothing that one both did and did not.
And here is where I will argue Bell-Lundy did this right. The overlap of things one ‘did do’ and ‘didn’t do’ must be empty. Do not be fooled by there being area in common in the overlap. One thing Venn Diagrams help us establish are the different kinds of things we are studying, and to work out whether that kind of thing can have any examples. And if the set of things in your life that you regret is empty — well! Is it not “living your best life”, as the caption advances, to have nothing one regrets doing, and nothing one regrets not doing? Thus I say to you the jury of readers, Sandra Bell-Lundy has correctly used the Venn Diagram form to make a “No Regrets” art.
That said, I can’t explain why the protagonist on the left is slumping and looking depressed. I suppose we have to take that she hasn’t lived her best life, but does have information about what might have been.
Jeff Mallet’s Frazz for the 1st starts a string of mathematics class jokes. Here is one about story problems, particularly ones about pricing apples and groups of apples. I don’t know whether apples are used as story problem examples. They seem like good example objects. They’re reasonably familiar. A person can have up to several dozen of them without it being ridiculously many. (Count a half-bushel of apples sometime.) You can imagine dividing them among people or tasks. You can even imagine halving and quartering them without getting ridiculous. Great set of traits. But the kid has overlooked that if Mrs Olsen wanted the price of an apple she would just look at the price sign.
(Every time I’m at the market I mean to check the apple prices, and I do, and I forget the total on the way out. I mention because I live in the same area as Jef Mallet. So there is a small but not-ridiculous chance he and I have bought apples from the same place. If he has a strip mentioning the place with the free coffee, popcorn, and gelato samples I’ll know to my satisfaction.)
Jeff Mallet’s Frazz for the 2nd has a complaint about having to show one’s work. But as with apple prices, we don’t really care whether someone has the right answer. We care whether they have the right method for finding an answer. Or, better, whether they have a method that could plausibly find the right answer, and an idea of how to check whether they did get it. This is why it’s worth, for example, working out a rough expected answer before doing a final calculation.
The talk about flight paths reminds me of a story passed around sci.space.history back in the day. The story is about development of the automatic landing computers used for the Apollo Missions. The guidance computers were programmed to get the lunar module from this starting point to a final point on the lunar surface. This turns into a question of polynomial interpolation. That’s coming up with a curve that fits some data points, particularly, the positions and velocities the last couple times those were known plus the intended landing position. You can always find a polynomial that passes smoothly through a finite bunch of data points. That’s not hard. But, allegedly, the guidance computer would project paths where the height above the lunar surface was negative for a while. Numerically, there’s nothing wrong with a negative number. It’s just got some practical problems, as the earliest Apollo missions were before any subway tunnels could be built.
Jeff Mallet’s Frazz for the 3rd continues the protest against showing one’s work. I do like the analogy of arithmetic skills for mathematics being like spelling skills for writing. You can carry on without these skills, for either mathematics or writing. But knowing them makes your life easier. And enjoying these building-block units foreshadows enjoying the whole. But yeah, addition and multiplication tables can look like tedium if you don’t find something at least a little thrilling in how, say, 9 times 7 is 63.
Three of these essays in a row now that Jef Mallett’s Frazz has done something worth responding to. You know, the guy lives in the same metro area. He could just stop in and visit sometime. There’s a pinball league in town and everything. He could view it as good healthy competition.
Bill Hinds’s Cleats for the 1st is another instance of the monkeys-on-typewriters metaphor. The metaphor goes back at least as far as 1913, when Émile Borel wrote a paper on statistical mechanics and the reversibility problem. Along the way it was worth thinking of the chance of impossibly unlikely events, given enough time to happen. Monkeys at typewriters formed a great image for a generator of text that knows no content or plan. Given enough time, this random process should be able to produce all the finite strings of text, whatever their content. And the metaphor’s caught people’s fancy I guess there’s something charming and Dadaist about monkeys doing office work. Borel started out with a million monkeys typing ten hours a day. Modern audiences sometimes make this an infinite number of monkeys typing without pause. This is a reminder of how bad we’re allowing pre-revolutionary capitalism get.
Sometimes it’s cut down to a mere thousand monkeys, as in this example. Often it’s Shakespeare, but sometimes it’s other authors who get duplicated. Dickens seems like a popular secondary choice. In joke forms, the number of monkeys and time it would take to duplicate something is held as a measure of the quality of the original work. This comes from people who don’t understand. Suppose the monkeys and typewriters are producing truly random strings of characters. Then the only thing that affects how long it takes them to duplicate some text is the length of the original text. How good the text is doesn’t enter into it.
Jef Mallett’s Frazz for the 1st is about the comfort of knowing about things one does not know. And that’s fine enough. Frazz cites Fermat’s Last Theorem as a thing everyone knows of but doesn’t understand. And that choice confuses me. I’m not sure what there would be to Fermat’s Last Theorem that someone who had heard of it would not understand. The basic statement of it — if you have three positive whole numbers a, b, and c, then there’s no whole number n larger than 2 so that equals — has it.
But “understanding” is a flexible concept. He might mean that people don’t know why the Theorem is true. Fair enough. Andrew Wiles and Richard Taylor’s proof is a long thing that goes deep into a field of mathematics that even most mathematicians don’t study. Why it should be true can be an interesting question, and one that’s hard to ever satisfyingly answer. What is the difference between a proof that something is true and an explanation for why it’s true? And before you say there’s not one, please consider that many mathematicians do experience a difference between seeing something proved and understanding why something is true.
And Frazz might also mean that nobody knows what use Fermat’s Last Theorem is. This is a fair complaint too. I’m not aware offhand of any interesting results which follow from its truth, nor of anything neat that would come about had it been false. It’s just one of those things that happens to be true, and that we’ve found to be pretty, perhaps because it is easy to ask whether it’s true and hard to answer. I don’t know.
Morrie Turner’s Wee Pals for the 2nd has a kid looking for a square root. We all have peculiar hobbies. His friends speak of it as though it’s a lost physical object. This is a hilarious misunderstanding until it strikes you that we speak about stuff like square roots “existing”. Indeed, the language of mathematics would be trashed if we couldn’t speak about numerical constructs “existing” somewhere to be “found”. But try to put “four” in a box and see what you get. That we mostly have little trouble understanding what we mean by showing some mathematical construct exists, and what we hope to do by looking for it, suggests we roughly know what we mean by the phrases. All right then; what is that, in terms a kid could understand?
There are many ways to numerically compute a square root, if you have to do it by hand and it isn’t a perfect square. My preference is for iterative methods, in which you start with a rough guess and try to improve things. One good enough method for we call the Babylonian method, reflecting how old we think it is. Start with your number S whose square root you want. And start with a number x0, a first guess for what the square root is. This can be anything. The great thing about iterative methods is even if you start with a garbage answer, you get to a good answer soon enough. Still, if you have a suspicion of what the square root should be, start there.
Your first iteration, the first guess for a better answer, is to calculate the number . Typically, x1 will be closer to the square root of S than will x0 be. And in any case, we can get closer still. Use x1 to calculate a new number. This is . And then x3 and x4 and x5 and so on. In theory, you never finish; you’re stuck finding an infinitely long sequence of better approximations to the square root. In practice, you finish; you find that you’re close enough to the square root. Well, the square root of a whole number is either a whole number (if it was a perfect square to start) or is an irrational number. You were going to stop on an approximation sooner or later.
The method requires doing division. Long division, too, after the first couple steps. I don’t know a way around that which doesn’t divert into something less pleasant, such as logarithms and exponentials. Or maybe into trigonometric functions. This can be tedious to do by hand. Great thing, though, is if you make a mistake? That’s kind of all right. The next iteration will (usually) correct for it. That’s the glory of iterative methods. They tend to be forgiving of numerical error, whatever its source. Another iteration reduces, or even eliminates, the mistake of the previous iteration.
For more of these Reading the Comics posts please follow this link. If you’re only interested in Reading the Cleats strips, please use this link instead. But Cleats is a new tag this essay, so for now, there aren’t others. If you’re hoping to see all my Reading the Comics posts about Frazz, try this link. If you’d like more of my essays which mention Wee Pals, you can use this link. And if you’d like more Reading the Comics posts that mention Harley, use this link. That’s another new tag, but I believe Dan Thompson is still making new examples of the strip. So it may appear again.
The last half of last week was not entirely the work of Chuckle Brothers and Saturday Morning Breakfast Cereal. It seemed like it, though. Let’s review.
Patrick Roberts’s Todd the Dinosaur for the 28th is a common sort of fear-of-mathematics joke. In this case the fear of doing arithmetic even when it is about something one would really like to know. I think the question got away from Todd, though. If they just wanted to know whether they had enough money, well, they need twelve dollars and have seven. Subtracting seven from twelve is only needed if they want to know how much more they need. Which they should want to know, but wasn’t part of the setup.
Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 28th uses mathematics as the sine qua non of rocket science. As in, well, the stuff that’s hard and takes some real genius to understand. It’s not clear to me that the equations are actually rocket science. There seem to be a shortage of things in exponentials to look quite right to me. But I can’t zoom in on the art, so, who knows just what might be in there.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th is a set theory joke. Or a logic joke, anyway. It refers to some of the mathematics/logic work of Bertrand Russell. Among his work was treating seriously the problems of how to describe things defined in reference to themselves. These have long been a source of paradoxes, sometimes for fun, sometimes for fairy-tale logic, and sometimes to challenge our idea of what we mean by definitions of things. Russell made a strong attempt at describing what we mean when we describe a thing by reference to itself. The iconic example here was the “set of all sets not members of themselves”.
Russell started out by trying to find some way to prove Georg Cantor’s theorems about different-sized infinities wrong. He worked out a theory of types, and what kinds of rules you can set about types of things. Most mathematicians these days prefer to solve the paradox with a particular organization of set theory. But Russell’s type theory still has value, particularly as part of the logic behind lambda calculus. This is an approach to organizing relationships between things that can do wonderful things, including in computer programming. It lets one write code that works extremely efficiently and can never be explained to another person, modified, or debugged ever. I may lack the proper training for the uses I’ve made of it.
Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 29th is a lottery joke. It does happen that more than one person wins a drawing; sometimes three or even four people do, for the larger prizes. The chance that there’s a million winners? Frightfully unlikely unless something significant went wrong with the lottery mechanism.
So what are the chances of a million lottery winners? If I’m not mistaken the only way to do this is to work out a binomial distribution. The binomial distribution is good for cases where you have many attempts at doing a thing, where each thing can either succeed or fail, and the likelihood of success or failure is independent of all the other attempts. In this case each lottery ticket is an attempt; it winning is success and it losing is failure. Each ticket has the same chance of winning or losing, and that chance doesn’t depend on how many wins or losses there are. What is that chance? … Well, if each ticket has one chance in a million of winning, and there are a million tickets out there, the chance of every one of them winning is about one-millionth raised to the millionth power. Which is so close to zero it might as well be nothing. … And yet, for all that it’s impossible, there’s not any particular reason it couldn’t happen. It just won’t.
Jef Mallet’s Frazz for the 29th is a less dire take on what-you-learned-this-year. In this case it’s trivia, but it’s a neat sort of trivia. Once you understand how it works you can understand how to make all sorts of silly little divisibility rules. The threes rule — and the nines rule — work by the same principle. Suppose you have a three-digit number. Let me call ‘a’ the digit in the hundreds column, ‘b’ the digit in the tens column, and ‘c’ the digit in the ones column. Then the number is equal to . And, well, that’s equal to . Which is . 99 times any whole number is a multiple of 9, and also of 3. 9 times any whole number is a multiple of 9, and also of 3. So whether the original number is divisible by 9, or by 3, depends on whether is. And that’s why adding the digits up tells you whether a number is a whole multiple of three.
This has only proven anything for three-digit numbers. But with that proof in mind, you probably can imagine what the proof looks like for two- or four-digit numbers, and would believe there’s one for five- and for 500-digit numbers. Or, for that matter, the proof for an arbitrarily long number. So I’ll skip actually doing that. You can fiddle with it if you want a bit of fun yourself.
Also maybe it’s me, or the kind of person who gets into mathematics. But I find silly little rules like this endearing. It’s a process easy to understand that anyone can do and it tells you something not obvious from when you start. It feels like getting let in on a magic trick. That seems like the sort of thing that endears people to mathematics.
Mike Thompson’s Grand Avenue for the 29th is trying to pick its fight with me again. I can appreciate someone wanting to avoid kids losing their mathematical skills over summer. It’s just striking how Thompson has consistently portrayed their grandmother as doing this in a horrible, joy-crushing manner.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 29th gets into a philosophy-of-mathematics problem. Also a pure philosophy problem. It’s a problem of what things you can know independently of experience. There are things it seems as though are true, and that seem independent of the person who is aware of them, and what culture that person comes from. All right. Then how can these things be relevant to the specifics of the universe that we happen to be in just now? If ‘2’ is an abstraction that means something independent of our universe, how can there be two books on the table? There’s something we don’t quite understand yet, and it’s taking our philosophers and mathematicians a long while to work out what that is.
For a while I thought this essay would include only the mathematically-themed strips which Comic Strip Master Command sent out through to June 26th, which is picking up the nickname Stitch Day (for 6-26, the movie character’s experiment number). And then I decided some from last Sunday weren’t on-point enough (somehow), and there were enough that came later in the week that I couldn’t do a June 26th Only edition. Which is my longwinded way of saying this one doesn’t have a nonsense name. It just has a name that’s only partially on point.
Mike Baldwin’s Cornered for the 26th is the Rubik’s Cube/strange geometry joke for the week. It seems to me I ought to be able to make some link between the number of various ways to arrange a Rubik’s Cube — which pieces can and which ones cannot be neighbors to a red piece, say, no matter how one scrambles the cube — and the networking between people that you can get from an office where people have to see each other. But I’m not sure that I can make that metaphor work. I’m blaming the temperature, both mine (I have a cold) and the weather’s (it’s a heat wave).
Mark Leiknes’s Cow and Boy for the 26th makes literal the trouble some people have with the phrase “110 percent”. Read uncharitably, yes, “110 of a hundred” doesn’t make sense, if 100 percent is all that could conceivably be of the thing. But if we can imagine, say, the number of cars passing a point on the highway being 90 percent of the typical number, surely we can imagine the number of cars also being 110 percent. To give an example of why I can’t side with pedants in objecting to the phrase.
Jef Mallett’s Frazz for the 26th is just itching for a fight. From me and from the Creative Writing department. Yes, mathematics rewards discipline. All activities do. At the risk of making a prescription: if you want to do something well, spend time practicing the boring parts. For arithmetic, that’s times tables and regrouping calculations and factoring and long division. For writing, that’s word choice and sentence structure and figuring how to bring life to describing dull stuff. Do the fun stuff too, yes, but because it is fun. Getting good at the boring stuff makes you an expert. When you discover that the boring stuff is also kinda fun, you will do the fun stuff masterfully.
But to speak of mathematics as pursuing a single right answer — well, perhaps. In an elementary-school problem there is typically just the one right answer, and the hope is that students learn how to get there efficiently. But if the subject is something well-worn, then there are many ways to do any problem. All are legitimate and the worst one can say of a method is maybe it’s not that efficient, or maybe it’s good here but doesn’t generally work. If the subject is on the edge of what mathematics we know, there may be only one way to get there. But there are many things to find, including original ways to understand what we have already found. To not see that mathematics is creative is to not see mathematics. Or, really, any field of human activity.
Samson’s Dark Side of the Horse for the 27th edges up to being the anthropomorphic numerals joke for the week. I need a good name for this sort of joke about mathematical constructs made tangible, even if they aren’t necessarily characters.
So. A “graph” is a thing that’s turned up in my A To Z serieses. In this context a graph is a collection of points, called “vertices”, and a collection of “edges” that connect vertices. Often the vertices represent something of interest and the edges ways to turn one thing into another. Sometimes the edges are the thing of interest and the vertices are just there to be manipulated in some way by edges. It’s a way to make visual the studying of how stuff is connected, and how things can pass from one to another.
A “stochastic process” is about random variables. Random variables are some property about a system. And you know some things about that variable’s value. You know maybe the range of possible values it could have. You know whether some values are more likely than others. But you do not know what the value is at any particular moment. Consider, say, the temperature outside where you live at a particular time of day. You may have no idea what that is. But you can say, for example, whether today it is more likely to be 90 degrees Fahrenheit or 60 degrees Fahrenheit or 20 degrees Fahrenheit. For a stochastic process we have some kind of index. We can say, for example, which values of temperature are more likely today, the 1st of July, and which ones will be more likely the 1st of August, and which ones will be less likely the 1st of December. Calling it a “process”, to my intuition, makes it sound like we expect something to happen that causes the likelihood of some temperatures to change. And many processes are time-indexed. They study problems where something interesting changes in time, predictable in aggregate but not in detail.
So a graph like this, representing a stochastic process, is a shorthand. Each vertex is a state that something might be in. Each edge is a way to get from one state to another when — something — happens. Doesn’t matter what thing.
A “drunk walk”, or as it’s known to tenderer writers a “random walk”, is a term of art. Not a deep one. It’s meant to evoke the idea of a severely drunk person who yes, can move, but has no control over which way. Thus he wanders around, reaching any point only by luck. Many things look like random walks, in which there is no overall direction, just an unpredictable shuffling around. A drunk walk on this graph would be, well, start at any of the vertices. Then follow edges, chosen randomly. If you start at the uppermost point of the triangle on top, for example, there’s two places to go on the second step: the lower-left or the center-right vertex on the upper triangle. Suppose you go to the center-right vertex. On the next step, you might go right back where you started. You might go to the lower-left vertex on the triangle. You might drop down that bridge to the top of that quadrilateral. And so on, for another step.
Do that some presumably big number of times. Where are you? … Anywhere, of course. But are there vertices you’re more likely to be on? Ones you’re less likely to be on? How does the shape of the graph affect that likelihood? How does how long you spend walking affect that? These tell us things about the process, and are why someone would draw this graph and talk about a random walk on it.
My week got busier than I imagined, but it was in ways worthwhile. I apologize for running late, and for not having an essay I meant to put up here this week. But I should be back to something more normal next week. I keep saying that. Also, for what seems like a rarity, all the strips for this essay are comic strips. No panels. That won’t last, I know.
Johnny Hart’s Back to B.C. for the 14th features arithmetic as a demonstration of The Smartest Man in the World’s credentials. I understand using a bit of arithmetic as a quick check that someone has any intelligence at all. It seems to me that checking “two plus two” is more common than “one plus one”, and either is more common than, say, “one plus two” or “three plus five” or anything. I’m curious why that is, though. Might one plus one just seem too simple? Or is it the bias against odd numbers and feeling that two plus two is somehow more balanced? If only there were some smart person I could ask.
Jef Mallett’s Frazz for the 14th has a blackboard full of arithmetic as the icon of “doing a lot of school work”. Can’t say it’s age-inappropriate or anything. It’s just an efficient way to show a lot of work that’s kind of tiring to do has been done. … Also somehow one of the commenters didn’t understand the use of ‘flag’ as meaning to lose energy or enthusiasm. Huh.
Jef Mallett’s Frazz for the 15th is a percentages joke, built on confusion between how to go from percentages to fractions and back again. Must say that I had thought 50 percent was tied well enough to one-half in ordinary language (or in phrases like splitting something fifty-fifty) that someone wouldn’t be confused by that. But everyone does miss some obvious things.
Mark Pett’s Lucky Cow for the 16th is a probability strip. It is based on what seems obvious, that the fact of any person’s existing is an incredibly unlikely event. We can imagine restarting the universe, and letting it all develop again. And we’re forced to conclude there are so many other ways that galaxies might form and stars might come into being and planets might form and life might develop and evolution might proceed and people might meet and children might be born, and only one way that gets us here. So the chance of any of us existing is impossibly tiny. This is all consistent with the “frequentist” idea of what probability means. In that, we say the probability of a thing happening is all the ways that it could happen divided by all the ways that something could happen. (There are a bunch of technical points to go along with this.)
But there are a lot of buried assumptions in there. Many of them seem reasonable. For example: could the universe unfold any differently? It seems obvious that, for example, the radius of the Earth’s orbit around the sun is arbitrary and might be anything in a band that could support life. And, surely, if the year had more or fewer days to it all human history would be different. But then this seems obvious: drop a bunch of short needles across a set of parallel straight lines. The number of needles that cross any of those lines should be arbitrary and unpredictable. Except that it is predictable; there’s a well-known formula that says how many of those needles have to cross those lines. The prediction can be lousy for a handful of needles. For millions of needles, though, it’ll be dead on. The universe won’t make sense any other way.
I can’t go so far as to say that it’s impossible for a universe to exist without me existing and just as I am. That seems egotistical. Even the needle-drop talk has room for variations on the universe. In ten million needle drops, one needle crossing more or less would not be an implausible difference. Ten or a thousand needles falling differently wouldn’t stand out. But, then, after enough needle drops? … If infinitely many needles dropped, I could say exactly what percentage of them crossed lines. (I am speaking so very casually about very difficult technical points. Please pretend I have clear answers for them.) There are deep philosophical questions about the idea of “other universes” that we have to ask if we want to take the subject seriously. But there are deep mathematical questions too.
Bob Shannon’s Tough Town for the 16th is more or less the anthropomorphized Roman Numerals joke for the week. I don’t know that there’s a strong consensus about why X was used to represent “ten”. Likely it’s impossible to prove any explanation is right. But X has settled into meaning ten, and to serve a host of other uses in typography and in symbols. Some of them are likely connected. Some are probably just coincidence.
Once again the name of this Reading the Comics edition has nothing to do with any of the strips. I’m just aware that Weezer’s cover of Africa is quite popular right now and who am I to deny people things they want? (I like the cover, but it’s not different enough for me to feel satisfied by it. I tend to like covers that highlight something minor in the original, or that go in a strange direction. Shifting a peppy song into a minor key doesn’t count anymore. But bear in mind, I’m barely competent at listening to music. Please now enjoy my eight hours of early electronica in which various beeps and whistles are passed off as music.)
Samson’s Dark Side of the Horse for the 3rd is the Roman numerals joke for the week. And a welcome return for Dark Side of the Horse. It feels like it’s been gone a while. I wouldn’t try counting by Roman numerals to lull myself to sleep; it seems like too much fussy detail work. But I suppose if you’ve gotten good at it, it’s easy.
Jef Mallett’s Frazz for the 3rd builds on removing statistics from their context. It’s a common problem. It’s possible to measure so very many things. Without a clear idea of what we should expect as normal the measurement doesn’t tell us much. And it can be hard to know what the right context for something even is. Let me deconstruct Caulfield’s example. We’re supposed to reflect on and consider that 40% of all weekdays are Monday and Friday too. But it’s not only weekdays that people work. Even someone working a Sunday might take a sick day. Monday and Friday are a bit over 28% of the whole week. But more people do work Monday-to-Friday than do Saturdays and Sundays, so the Sunday sick day is surely rarer than the Monday. So even if we grant Caulfield’s premise, what does it tell us?
Jason Chatfield’s Ginger Meggs for the 3rd is a bit of why-learn-mathematics propaganda. Megg’s father has a good answer. But it does shift the question back one step. Also I see in the top row that Meggs has one of those comic-strip special editions where the name of the book is printed on the back cover instead. (I’m also skeptical of the photo and text layout on the newspaper Megg’s father is reading. But I don’t know the graphic design style of Australian, as opposed to United States, newspapers.)
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 3rd may belong on some philosopher’s Reading the Comics blog instead. No matter. There’s some mathematical-enough talk going on here. There’s often many ways to approach the same problem. For example, approaching a system as a handful of items. Or as a huge number of them. Or as infinitely many things. Or as a continuum of things. There are advantages each way. A handful of things, for example, we can often model as interactions between pairs of things. We can model a continuum as a fluid. A vast number of things can let one’s computer numerically approximate a fluid. Or infinitely many particles if that’s more convenient.
To describe all these different models as sharing an “ontology-space” is good mathematical jargon too. In this context the “-space” would mean the collection of all these things that are built by the same plan but with different values of whichever parameter matters.
Bud Blake’s Tiger for the 6th of August, 1965 features Einstein’s famous equation. I suppose it’s showing how well-informed Julian is, that he knows and can present such a big result. There is beauty in mathematics (and physics). Mathematicians (and physicists) find the subject beautiful to start with, and try to find attractive results. I’m curious what the lay reader makes of mathematical symbols, though, just as pieces of art. I remember as a child finding this beauty in a table of integrals in the front of one of my mother’s old college textbooks. All those parallel rows of integral symbols drew me in though nothing I’d seen in mathematics had prepared me to even read it. I still find that beautiful, but I can’t swear that I would even if I hadn’t formed that impression early in life. Are lay and professional readers’ views of mathematical-expression beauty similar? How are they different?
I learn, from reading not-yet-dead Usenet group rec.arts.comics.strips, that Rick Stromoski is apparently ending the comic Soup To Nutz. This is sad enough. But worse, GoComics.com has removed all but the current day’s strip from its archives. I had trusted that GoComics.com links were reliable in a way that Comics Kingdom and Creators.com weren’t. Now I learn that maybe I need to include images of the comics I review and discuss here lest my essays become unintelligible in the future? That’s not a good sign. I can do it, mind you. I just haven’t got started. You’ll know when I swing into action.
Norm Feuti, of Retail, still draws Sunday strips for Gil. They’re to start appearing on GoComics.com soon, and I can talk about them from my regular sources after that. But for now I follow the strip on Twitter. And last Sunday he posted this one.
It’s sort of a protesting-the-problem question. It’s also a reaction a lot of people have to “explain how you found the answer” questions. In a sense, yeah, the division shows how the answer was found. But what’s wanted — and what’s actually worth learning — is to explain why you did this calculation. Why, in this case, 216 divided by 8? Why not 216 times 8? Why not 8 divided by 216? Why not 216 minus 8? “How you found your answer” is probably a hard question to make interesting on arithmetic, unfortunately. If you’re doing a long sheet of problems practicing division, it’s not hard to guess that dividing is the answer. And that it’s the big number divided by the small. It can be good training to do blocks of problems that use the same approach, for the same reason it can be good training to focus on any exercise a while. But this does cheat someone of the chance to think about why one does this rather than that.
Patrick Roberts’s Todd the Dinosaur for the 11th has mathematics as the thing Todd’s trying to get out of doing. (I suppose someone could try to argue the Y2K bug was an offshoot of mathematics, on the grounds that computer science has so much to do with mathematics. I wouldn’t want to try defending that, though.) I grant that most fraction-to-decimal conversion problems hit that sweet spot of being dull, tedious, and seemingly pointless. There’s some fun decimal expansions of fractions. The sevenths and the elevenths and 1/243 have charm to them. There’s some kid who’ll become a mathematician because at the right age she was told about . 3/16th? Eh.
Mark Anderson’s Andertoons for the 11th is the Mark Anderson’s Andertoons for the week. I don’t remember seeing a spinny wheel like this used to introduce probability. It’s a good prop, though. I would believe in a class having it.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 11th is built on the Travelling Salesman Problem. It’s one of the famous unsolved and hard problems of mathematics. Weinersmith’s joke is a nice gag about one way to “solve” the problem, that of making it irrelevant. But even if we didn’t need to get to a collection of places efficiently mathematicians would still like to know good ways to do it. It turns out that finding the shortest (quickest, cheapest, easiest, whatever) route connecting a bunch of places is great problem. You can phrase enormously many problems about doing something as well as possible as a Travelling Salesman Problem. It’s easy conceptually to find the answer: try out all the possibilities and pick the best one. But if there’s more than a handful of cities, there are so many possible routes there’s no checking them all, not before you die of old age. We can do very well finding approximate answers, including by my specialization of Monte Carlo methods. In those you take a guess at an answer. Then make, randomly, a change. You’ll either have made things better or worse. If you’ve made it better, keep the change. If you’ve made it worse, usually you reject the change but sometimes you keep it. And repeat. In surprisingly little time you’ll get a really good answer. Maybe not the best possible, but a great answer for how straightforward setting it up was.
Computer science prof tells students one of the major problems in theoretical computer science is to prove P = NP; less than stellar student announces that P = 0 is a solution. https://t.co/xH9l8xkzzn
Dan Thompson’s Brevity for the 12th is a Rubik’s Cube joke. There’s not a lot of mathematics to that. But I do admire how Thompson was careful enough to draw a Rubik’s Cube that actually looks like the real article; it’s not just an isometric cube with thick lines partitioning it. Look at the corners of each colored sub-cube. I may be the only reader to notice this but I’m glad Thompson did the work.
Mason Mastroianni’s The Wizard of Id for the 12th gets Sir Rodney in trouble with the King for doing arithmetic. I haven’t read the comments on GoComics.com. I’d like to enter “three” as my guess for how many comments one would have to read before finding the “weapons of math instruction” joke in there.
Steve Moore’s In The Bleachers for the 13th features a story problem as a test of mental acuity. When the boxer can’t work out what the heck the trains-leaving-Penn-Station problem even means he’s ruled unfit to keep boxing. The question is baffling, though. As put, the second train won’t ever overtake the first. The question: did Moore just slip up? If the first train were going 30 miles per hour and the second 40 there would be a perfectly good, solvable question in this. Or was Moore slipping in an extra joke, making the referee’s question one that sounds like it was given wrong? Don’t know, so I’ll suppose the second.
Comic Strip Master Command spent most of February making sure I could barely keep up. It didn’t slow down the final week of the month either. Some of the comics were those that I know are in eternal reruns. I don’t think I’m repeating things I’ve already discussed here, but it is so hard to be sure.
Bill Amend’s FoxTrot for the 24th of February has a mathematics problem with a joke answer. The approach to finding the area’s exactly right. It’s easy to find areas of simple shapes like rectangles and triangles and circles and half-circles. Cutting a complicated shape into known shapes, finding those areas, and adding them together works quite well, most of the time. And that’s intuitive enough. There are other approaches. If you can describe the outline of a shape well, you can use an integral along that outline to get the enclosed area. And that amazes me even now. One of the wonders of calculus is that you can swap information about a boundary for information about the interior, and vice-versa. It’s a bit much for even Jason Fox, though.
Jef Mallett’s Frazz for the 25th is a dispute between Mrs Olsen and Caulfield about whether it’s possible to give more than 100 percent. I come down, now as always, on the side that argues it depends what you figure 100 percent is of. If you mean “100% of the effort it’s humanly possible to expend” then yes, there’s no making more than 100% of an effort. But there is an amount of effort reasonable to expect for, say, an in-class quiz. It’s far below the effort one could possibly humanly give. And one could certainly give 105% of that effort, if desired. This happens in the real world, of course. Famously, in the right circles, the Space Shuttle Main Engines normally reached 104% of full throttle during liftoff. That’s because the original specifications for what full throttle would be turned out to be lower than was ultimately needed. And it was easier to plan around running the engines at greater-than-100%-throttle than it was to change all the earlier design documents.
Matt Janz’s Out of the Gene Pool rerun for the 25th tosses off a mention of “New Math”. It’s referenced as a subject that’s both very powerful but also impossible for Pop, as an adult, to understand. It’s an interesting denotation. Usually “New Math”, if it’s mentioned at all, is held up as a pointlessly complicated way of doing simple problems. This is, yes, the niche that “Common Core” has taken. But Janz’s strip might be old enough to predate people blaming everything on Common Core. And it might be character, that the father is old enough to have heard of New Math but not anything in the nearly half-century since. It’s an unusual mention in that “New” Math is credited as being good for things. (I’m aware this strip’s a rerun. I had thought I’d mentioned it in an earlier Reading the Comics post, but can’t find it. I am surprised.)
Ah, yes, so, in the midst of feeling all proud that I’d gotten my Reading the Comics workflow improved, I went out to do my afternoon chores without posting the essay. I’m embarrassed. But it really only affects me looking at the WordPress Insights page. It publishes this neat little calendar-style grid that highlights the days when someone’s posted and this breaks up the columns. This can only unnerve me. I deserve it.
Tom Thaves’s Frank and Ernest for the 8th of February is about the struggle to understand zero. As often happens, the joke has a lot of truth to it. Zero bundles together several ideas, overlapping but not precisely equal. And part of that is the idea of “nothing”. Which is a subtly elusive concept: to talk about the properties of a thing that does not exist is hard. As adults it’s easy to not notice this anymore. Part’s likely because mastering a concept makes one forget what it took to understand. Part is likely because if you don’t have to ponder whether the “zero” that’s “one less than one” is the same as the “zero” that denotes “what separates the count of thousands from the count of tens in the numeral 2,038” you might not, and just assume you could explain the difference or similarity to someone who has no idea.
John Zakour and Scott Roberts’s Maria’s Day for the 8th has maria and another girl bonding over their hatred of mathematics. Well, at least they’re getting something out of it. The date in the strip leads me to realize this is probably a rerun. I’m not sure just when it’s from.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th proposes a prank based on mathematical use of the word “arbitrarily”. This is a word that appears a lot in analysis, and the strip makes me realize I’m not sure I can give a precise definition. An “arbitrarily large number”, for example, would be any number that’s large enough. But this also makes me realize I’m not sure precisely what joke Weinersmith is going for. I suppose that if someone were to select an arbitrarily large number they might pick 53, or a hundred, or million billion trillion. I suppose Weinersmith’s point is that in ordinary speech an arbitrarily made choice is one selection from all the possible alternatives. In mathematical speech an arbitrarily made choice reflects every possible choice. To speak of an arbitrarily large number is to say that whatever selection is made, we can go on to show this interesting stuff is true. We’d typically like to prove the most generically true thing possible. But picking a single example can be easier to prove. It can certainly be easier to visualize. 53 is probably easier to imagine than “every number 52 or larger”, for example.
Ted Shearer’s Quincy for the 16th of December, 1978 was rerun the 9th of February. It just shows Quincy at work on his mathematics homework, and considering dedicating it to his grandmother. Mathematics books have dedications, just as any other book does. I’m not aware of dedications of proofs or other shorter mathematics works, but there’s likely some. There’s often a note of thanks, usually given to people who’ve made the paper’s writers think harder about the subjects. But I don’t think there’s any reason a paper wouldn’t thank someone who provided “mere” emotional support. I just don’t have examples offhand.
Jef Mallet’s Frazz for the 9th looks like one of those creative-teaching exercises I sometimes see in Mathematics Education Twitter: the teacher gives answers and the students come up with story problems to match. That’s not a bad project. I’m not sure how to grade it, but I haven’t done anything that creative when I’ve taught. I’m sorry I haven’t got more to say about it since the idea seems fun.
Gordon Bess’s Redeye for the 30th of September, 1971 was rerun the 10th. It’s a bit of extremely long division and I don’t blame Pokey for giving up on that problem. Starting from 5,967,342 divided by 973 I’d say, well, that’s about six million divided by a thousand, so the answer should be near six thousand. I don’t think the last digits of 2 and 3 suggest anything about what the final digit should be, if this divides evenly. So the only guidance I have is that my answer ought to be around six thousand and then we have to go into actually working. It turns out that 973 doesn’t go into 5,967,342 a whole number of times, so I sympathize more with Pokey. The answer is a little more than 6,132.9311.
It wasn’t much of an increased workload, really. I mean, none of the comics required that much explanation. But Comic Strip Master Command donated enough topics to me last week that I have a second essay for the week. And here it is; sorry there’s no pictures.
Lorrie Ransom’s The Daily Drawing for the 18th is another name-drop of mathematics. I guess it’s easier to use mathematics as the frame for saying something’s just a “problem”. I don’t think of, say, identifying the themes of a story as a problem in the way that finding the roots of a quadratic is.