I think of myself as not a prescriptivist blogger. Here and on my humor blog I do what I feel like, and if that seems to work, I do more of it if I can. If I do enough of it, I try to think of a title, give up and use the first four words that kind of fit, and then ask Thomas K Dye for header art. If it doesn’t work, I drop it without mention. Apart from appealing for A-to-Z topics I don’t usually declare what I intend to do.
This feels different. One of the first things I fell into here, and the oldest hook in my blogging, is Reading the Comics. It’s mostly fun. But it is also work. 2020 is not a year when I am capable of expanding my writing work without bounds. Something has to yield, and my employers would rather it not be my day job. So, at least through the completion of the All 2020 Mathematics A-to-Z, I’ll just be reading the comics. Not Reading the Comics for posting here.
And this is likely a good time for a hiatus. There is much that’s fun about Reading the Comics. First is the comic strips, a lifelong love. Second is that they solve the problem of what to blog about. During the golden age of Atlantic City, there was a Boardwalk performer whose gimmick was to drag a trap along the seabed, haul it up, and identify every bit of sea life caught up in that. My schtick is of a similar thrill, with less harm required of the sea life.
But I have felt bored by this the last several months. Boredom is not a bad thing, of course. And if you are to be a writer, you must be able to write something competent and fresh about a topic you are tired of. Admitting that: I do not have one more sentence in me about kids not buying into the story problem. Or observing that yes, that is a blackboard full of mathematics symbols. Or that lotteries exist and if you play them infinitely many times strange conclusions seem to follow. An exercise that is tiring can be good; an exercise that is painful is not. I will put the painful away and see what I feel like later.
For the time being I figure to write only the A-to-Z essays. And, since I have them, to post references back to old A-to-Z essays. These recaps seemed to be received well enough last year. So why not repeat something that was fine when it was just one of many things?
And after all, the A-to-Z theme is still at heart hauling up buckets of sea life and naming everything in it. It’s just something that I can write farther ahead of deadline, but will not.
Thanks all for reading.
The Boardwalk performer would, if stumped, make up stuff. What patron was going to care if they went away ill-informed? It was a show. The performer just needed a confident air.
Let’s see if I can’t close out the first week of June’s comics. I’d rather have published this either Tuesday or Thursday, but I didn’t have the time to write my statistics post for May, not yet. I’ll get there.
One of Gary Larson’s The Far Side reprints for the 4th is one I don’t remember seeing before. The thing to notice is the patient has a huge right brain and a tiny left one. The joke is about the supposed division between left-brained and right-brained people. There are areas of specialization in the brain, so that the damage or destruction of part can take away specific abilities. The popular imagination has latched onto the idea that people can be dominated by specialties of the either side of the brain. I’m not well-versed in neurology. I will hazard the guess that neurologists see “left-brain” and “right-brain” as amusing stuff not to be taken seriously. (My understanding is the division of people into “type A” and “type B” personalities is also entirely bunk unsupported by any psychological research.)
Bud Blake’s Tiger rerun for the 6th has Tiger complaining about his arithmetic homework. And does it in pretty nice form, really, doing some arithmetic along the way. It does imply that he’s starting his homework at 1 pm, though, so I guess it’s a weekend afternoon. It seems like rather a lot of homework for that age. Maybe he’s been slacking off on daily work and trying to make up for it.
John McPherson’s Close To Home for the 6th has a cheat sheet skywritten. It’s for a geometry exam. Any subject would do, but geometry lets cues be written out in very little space. The formulas are disappointingly off, though. We typically use ‘r’ to mean the radius of a circle or sphere, but then would use C for its circumference. That would be . The area of a circle, represented with A, would be . I’m not sure what ‘Vol.C’ would mean, although ‘Volume of a cylinder’ would make sense … if the next line didn’t start “Vol.Cyl”. The volume of a circular cylinder is , where r is the radius and h the height. For a non-circular cylinder, it’s the area of a cross-section times the height. So that last line may be right, if it extends out of frame.
Granted, though, a cheat sheet does not necessarily make literal sense. It needs to prompt one to remember what one needs. Notes that are incomplete, or even misleading, may be all that one needs.
Bob Weber Jr’s Slylock Fox for the 1st of June sees Reeky Rat busted for speeding on the grounds of his average speed. It does make the case that Reeky Rat must have travelled faster than 20 miles per hour at some point. There’s no information about when he did it, just the proof that there must have been some time when he drove faster than the speed limit. One can find loopholes in the reasoning, but, it’s a daily comic strip panel for kids. It would be unfair to demand things like proof there’s no shorter route from the diner and that the speed limit was 20 miles per hour the whole way.
Ted Shearer’s Quincy for the 1st originally ran the 7th of April, 1981. Quincy and his friend ponder this being the computer age, and whether they can let computers handle mathematics.
Jef Mallett’s Frazz for the 2nd has the characters talk about how mathematics offers answers that are just right or wrong. Something without “subjective grading”. It enjoys that reputation. But it’s not so, and that’s obvious when you imagine grading. How would you grade an answer that has the right approach, but makes a small careless error? Or how would you grade an approach that doesn’t work, but that plausibly could?
And how do you know that the approach wouldn’t work? Even in non-graded mathematics, we have subjectivity. Much of mathematics is a search for convincing arguments about some question. What we hope to be convinced of is that there is a sound logical argument making the same conclusions. Whether the argument is convincing is necessarily subjective.
Yes, in principle, we could create a full deductive argument. It will take forever to justify every step from some axiom or definition or rule of inference. And even then, how do we know a particular step is justified? It’s because we think we understand what the step does, and how it conforms to one (or more) rule. That’s again a judgement call.
(The grading of essays is also less subjective than you might think if you haven’t been a grader. The difference between an essay worth 83 points and one worth 85 points may be trivial, yes. But you will rarely see an essay that reads as an A-grade one day and a C-grade the next. This is not to say that essay grading is not subject to biases. Some of these are innocent, such as the way the grader’s mood will affect the grade. Or how the first several papers, or the last couple, will be less consistently graded than the ones done in the middle of the project. Some are pernicious, such as under-rating the work done by ethnic minority students. But these biases affect the way one would grade, say, the partial credit for an imperfectly done algebra problem too.)
Mark Anderson’s Andertoons for the 3rd is the Mark Anderson’s Andertoons for the week. I could also swear that I’ve featured it here before. I can’t find it, if I have discussed this strip before. I may not have. Wavehead’s observing the difference between zero as an additive identity and its role in multiplication.
Ryan Pagelow’s Buni for the 3rd fits into the anthropomorphic-numerals category of joke. It’s really more of a representation of the year as the four horsemen of the Apocalypse.
Doug Savage’s Savage Chickens for the 3rd has, as part of animal facts, the assertion that “llamas have basic math skills”. I don’t know of any specific research on llama mathematics skills. But animals do have mathematics skills. Often counting. Some amount of reasoning. Social animals often have an understanding of transitivity, as well, especially if the social groups have a pecking order.
This is the slightly belated close of last week’s topics suggested by Comic Strip Master Command. For the week we’ve had, I am doing very well.
Werner Wejp-Olsen’s Inspector Danger’s Crime Quiz for the 25th of May sees another mathematician killed, and “identifying” his killer in a dying utterance. Inspector Danger has followed killer mathematicians several times before: the 9th of July, 2012, for instance. Or the 4th of July, 2016, for a case so similar that it’s almost a Slylock Fox six-differences puzzle. Apparently realtors and marine biologists are out for mathematicians’ blood. I’m not surprised by the realtors, but hey, marine biology, what’s the deal? The same gimmick got used the 15th of May, 2017, too. (And in fairness to the late Wejp-Olsen, who could possibly care that similar names are being used in small puzzles used years apart? It only stands out because I’m picking out things that no reasonable person would notice.)
Jim Meddick’s Monty for the 25th has the title character inspired by the legend of genius work done during plague years. A great disruption in life is a great time to build new habits, and if Covid-19 has given you the excuse to break bad old habits, or develop good new ones, great! Congratulations! If it has not, though? That’s great too. You’re surviving the most stressful months of the 21st century, I hope, not taking a holiday.
Anyway, the legend mentioned here includes Newton inventing Calculus while in hiding from the plague. The actual history is more complicated, and ambiguous. (You will not go wrong supposing that the actual history of a thing is more complicated and ambiguous than you imagine.) The Renaissance Mathematicus describes, with greater authority and specificity than I could, what Newton’s work was more like. And some of how we have this legend. This is not to say that the 1660s were not astounding times for Newton, nor to deny that he worked with a rare genius. It’s more that we are lying to imagine that Newton looked around, saw London was even more a deathtrap than usual, and decided to go off to the country and toss out a new and unique understanding of the infinitesimal and the continuum.
Mark Anderson’s Andertoons for the 27th is the Mark Anderson’s Andertoons for the week. One of the students — not Wavehead — worries that a geometric ray, going on forever, could endanger people. There’s some neat business going on here. Geometry, like much mathematics, works on abstractions that we take to be universally true. But it also seems to have a great correspondence to ordinary real-world stuff. We wouldn’t study it if it didn’t. So how does that idealization interact with the reality? If the ray represented by those marks on the board goes on to do something, do we have to take care in how it’s used?
Comic Strip Master Command wanted to give me a break as I ready for the All 2020 A-to-Z. I appreciate the gesture, especially given the real-world events of the past week. I get to spend this week mostly just listing appearances, even if they don’t inspire deeper thought.
Gordon Bess’s vintage Redeye for the 24th has one of his Cartoon Indians being lousy at counting. Talking about his failures at arithmetic, with how he doesn’t count six shots off well. There’s a modest number of things that people are, typically, able to perceive at once. Six can be done, although it’s easy for a momentary loss of focus to throw you off. This especially for things that have to be processed in sequence, rather than perceived all together.
Wulff and Morgenthaler’s WuMo for the 24th shows a parent struggling with mathematics, billed as part of “the terrible result of homeschooling your kids”. It’s a cameo appearance. It’d be the same if Mom were struggling with history or English. This is just quick for the comic strip reader to understand.
Andrés J. Colmenares’s Wawawiwa for the 25th sets several plants in a classroom. They’re doing arithmetic. This, too, could be any course; it just happens to be mathematics.
Sam Hurt’s Eyebeam for the 25th is built on cosmology. The subject is a blend of mathematics, observation, and metaphysics. The blackboard full of mathematical symbols gets used as shorthand for describing the whole field, not unfairly. The symbols as expressed don’t come together to mean anything. I don’t feel confident saying they don’t mean anything, though.
This was a week of few mathematically-themed comic strips. I don’t mind. If there was a recurring motif, it was about parents not doing mathematics well, or maybe at all. That’s not a very deep observation, though. Let’s look at what is here.
Liniers’s Macanudo for the 18th puts forth 2020 as “the year most kids realized their parents can’t do math”. Which may be so; if you haven’t had cause to do (say) long division in a while then remembering just how to do it is a chore. This trouble is not unique to mathematics, though. Several decades out of regular practice they likely also have trouble remembering what the 11th Amendment to the US Constitution is for, or what the rule is about using “lie” versus “lay”. Some regular practice would correct that, though. In most cases anyway; my experience suggests I cannot possibly learn the rule about “lie” versus “lay”. I’m also shaky on “set” as a verb.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th shows a mathematician talking, in the jargon of first and second derivatives, to support the claim there’ll never be a mathematician president. Yes, Weinersmith is aware that James Garfield, 20th President of the United States, is famous in trivia circles for having an original proof of the Pythagorean theorem. It would be a stretch to declare Garfield a mathematician, though, except in the way that anyone capable of reason can be a mathematician. Raymond Poincaré, President of France for most of the 1910s and prime minister before and after that, was not a mathematician. He was cousin to Henri Poincaré, who founded so much of our understanding of dynamical systems and of modern geometry. I do not offhand know what presidents (or prime ministers) of other countries have been like.
Weinersmith’s mathematician uses the jargon of the profession. Specifically that of calculus. It’s unlikely to communicate well with the population. The message is an ordinary one, though. The first derivative of something with respect to time means the rate at which things are changing. The first derivative of a thing, with respect to time being positive means that the quantity of the thing is growing. So, that first half means “things are getting more bad”.
The second derivative of a thing with respect to time, though … this is interesting. The second derivative is the same thing as the first derivative with respect to time of “the first derivative with respect to time”. It’s what the change is in the rate-of-change. If that second derivative is negative, then the first derivative will, in time, change from being positive to being negative. So the rate of increase of the original thing will, in time, go from a positive to a negative number. And so the quantity will eventually decline.
So the mathematician is making a this-is-the-end-of-the-beginning speech. The point at which the the second derivative of a quantity changes sign is known as the “inflection point”. Reaching that is often seen as the first important step in, for example, disease epidemics. It is usually the first good news, the promise that there will be a limit to the badness. It’s also sometimes mentioned in economic crises or sometimes demographic trends. “Inflection point” is likely as technical a term as one can expect the general public to tolerate, though. Even that may be pushing things.
Julie Larson’s The Dinette Set rerun for the 21st fusses around words. Along the way Burl mentions his having learned that two negatives can make a positive, in mathematics. Here it’s (most likely) the way that multiplying or dividing two negative numbers will produce a positive number.
The end of last week offered just a few more comic strips, and some pretty casual mathematics content. Let me wrap that up.
Daniel Beyer’s Long Story Short for the 13th has the “math department lavatory” represented as a door labelled . It’s an interesting joke in that it reads successfully, but doesn’t make sense. To match the references to the commonly excreted substances they’d want .
Gary Larson’s The Far Side strips for the 14th includes the famous one of Albert Einstein coming so close to working out . The usual derivations for don’t start with that and then explore whether it makes sense, which is what Einstein seems to be doing here. Instead they start from some uncontroversial premises and find that they imply this business. Dimensional analysis would also let you know that, if c is involved, it’s probably to the second power rather than anything else.
But that doesn’t mean we can’t imagine Einstein assuming there must be a relationship between energy and mass, finding one that makes sense, and then finding a reason it’s that rather than something else. That’s a common enough pattern of mathematical discovery. Also, a detail I hadn’t noticed before, is that Einstein tried out , rejected it, and then tried it again. This is also a common pattern of discovery.
The past week had a fair number of comic strips mentioning some aspect of mathematics. One of them is, really, fairly slight. But it extends a thread in the comic strip that I like and so that I will feature here.
Sam Hurt’s Eyebeam for the 11th uses heaps of mathematical expressions, graphs, charts, and Venn diagrams to represent the concept of “data”. It’s spilled all over to represent “sloppy data”. Usually by the term we mean data that we feel is unreliable. Measurements that are imprecise, or that are unlikely to be reliable. Precision is, roughly, how many significant digits your measurement has. Reliability is, roughly, if you repeated the measurement would you get about the same number?
We’re accustomed in probability to thinking of the expectation value. This is the chance that something will happen, given some number N opportunities to happen, if at each opportunity it has the probability p of happening. Let me assume the probability is always the same number. If it’s not, our work gets harder, although it’s basically the same kind of work. But, then, the expectation value, the number of times we’d expect to see the thing happen, is N times p. Which, as Utahraptor points out, we can expect has to be at least 1 for any event, however unlikely, given enough chances. So it should be.
But, then, to take Utahraptor’s example: what is the probability that an immortal being never trips down the stairs? At least not badly enough to do harm? Why should we think that’s zero? It’s not as if there’s a physical law that compels someone to go to stairs and then to fall down them to their death. And, if there’s any nonzero chance of someone not dying this way? Then, if there are enough immortals, there’s someone who will go forever without falling down stairs.
That covers just the one way to die, of course. But the same reasoning holds for every possible way to die. If there’s enough immortals, there’s someone who would not die from falling down stairs and from never being struck by a meteor. And someone who’d never fall down stairs and never be struck by a meteor and never fall off a cliff trying to drop an anvil on a roadrunner. And so on. If there are infinitely many people, there’s at least one who’d avoid all possible accidental causes of death.
More. If there’s infinitely many immortals, then there are going to be a second and a third — indeed, an infinite number — of people who happen to be lucky enough to never die from anything. Infinitely many immortals die of accidents, sure, but somehow not all of them. We can’t even say that more immortals die of accidents than don’t.
My point is that probability gets really weird when you try putting infinities into it. Proceed with extreme caution. But the results of basic, incautious, thinking can be quite heady.
Bill Amend’s FoxTrot Classics for the 12th has Paige cramming for a geometry exam. Don’t cram for exams; it really doesn’t work. It’s regular steady relaxed studying that you need. That and rest. There is nothing you do that you do better for being sleep-deprived.
Olivia Jaimes’s Nancy for the 8th has Nancy and Sluggo avoiding mathematics homework. Or, “practice”, anyway. There’s more, though; Nancy and Sluggo are doing some analysis of viewing angles. That’s actual mathematics, certainly. Computer-generated imagery depends on it, just like you’d imagine. There are even fun abstract questions that can give surprising insights into numbers. For example: imagine that space were studded, at a regular square grid spacing, with perfectly reflective marbles of uniform size. Is there, then, a line of sight between any two points outside any marbles? Even if it requires tens of millions of reflections; we’re interested in what perfect reflections would give us.
Using playing cards as a makeshift protractor is a creative bit of making do with what you have. The cards spread in a fanfold easily enough and there’s marks on the cards that you can use to keep your measurements reasonably uniform. Creating ad hoc measurement tools like this isn’t mathematics per se. But making a rough tool is a first step to making a precise tool. And you can use reason to improve your estimates.
It’s not on-point, but I did want to share the most wondrous ad hoc tool I know of: You can use an analog clock hand, and the sun, as a compass. You don’t even need a real clock; you can draw the time on a sheet of paper and use that. It’s not a precise measure, of course. But if you need some help, here you go. You’ve got it.
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.
The past week was a light one for mathematically-themed comic strips. So let’s see if I can’t review what’s interesting about them before the end of this genially dumb movie (1940’s Hullabaloo, starring Frank Morgan and featuring Billie Burke in a small part). It’ll be tough; they’re reaching a point where the characters start acting like they care about the plot either, which is usually the sign they’re in the last reel.
Jenny Campbell’s Flo and Friends for the 26th is a joke about fumbling a bit of practical mathematics, in this case, cutting a recipe down. When I look into arguments about the metric system, I will sometimes see the claim that English traditional units are advantageous for cutting down a recipe: it’s quite easy to say that half of “one cup” is a half cup, for example. I doubt that this is much more difficult than working out what half of 500 ml is, and my casual inquiries suggest that nobody has the faintest idea what half of a pint would be. And anyway none of this would help Ruthie’s problem, which is taking two-fifths of a recipe meant for 15 people. … Honestly, I would have just cut it in half and wonder who’s publishing recipes that serve 15.
Ed Bickford and Aaron Walther’s American Chop Suey for the 28th uses a panel of (gibberish) equations to represent deep thinking. It’s in part of a story about an origami competition. This interests me because there is serious mathematics to be done in origami. Most of these are geometry problems, as you might expect. The kinds of things you can understand about distance and angles from folding a square may surprise. For example, it’s easy to trisect an arbitrary angle using folded squares. The problem is, famously, impossible for compass-and-straightedge geometry.
Origami offers useful mathematical problems too, though. (In practice, if we need to trisect an angle, we use a protractor.) It’s good to know how to take a flat, or nearly flat, thing and unfold it into a more interesting shape. It’s useful whenever you have something that needs to be transported in as few pieces as possible, but that on site needs to not be flat. And this connects to questions with pleasant and ordinary-seeming names like the map-folding problem: can you fold a large sheet into a small package that’s still easy to open? Often you can. So, the mathematics of origami is a growing field, and one that’s about an accessible subject.
Bill Holbrook’s On The Fastrack for the 2nd of May also talks about the use of x as a symbol. Curt takes eagerly to the notion that a symbol can represent any number, whether we know what it is or not. And, also, that the choice of symbol is arbitrary; we could use whatever symbol communicates. I remember getting problems to work in which, say, 3 plus a box equals 8 and working out what number in the box would make the equation true. This is exactly the same work as solving 3 + x = 8. Using an empty box made the problem less intimidating, somehow.
Dave Whamond’s Reality Check for the 2nd is, really, a bit baffling. It has a student asking Siri for the cosine of 174 degrees. But it’s not like anyone knows the cosine of 174 degrees off the top of their heads. If the cosine of 174 degrees wasn’t provided in a table for the students, then they’d have to look it up. Well, more likely they’d be provided the cosine of 6 degrees; the cosine of an angle is equal to minus one times the cosine of 180 degrees minus that same angle. This allows table-makers to reduce how much stuff they have to print. Still, it’s not really a joke that a student would look up something that students would be expected to look up.
… That said …
If you know anything about trigonometry, you know the sine and cosine of a 30-degree angle. If you know a bit about trigonometry, and are willing to put in a bit of work, you can start from a regular pentagon and work out the sine and cosine of a 36-degree angle. And, again if you know anything about trigonometry, you know that there are angle-addition and angle-subtraction formulas. That is, if you know the cosine of two angles, you can work out the cosine of the difference between them.
So, in principle, you could start from scratch and work out the cosine of 6 degrees without using a calculator. And the cosine of 174 degrees is minus one times the cosine of 6 degrees. So it could be a legitimate question to work out the cosine of 174 degrees without using a calculator. I can believe in a mathematics class which has that as a problem. But that requires such an ornate setup that I can’t believe Whamond intended that. Who in the readership would think the cosine of 174 something to work out by hand? If I hadn’t read a book about spherical trigonometry last month I wouldn’t have thought the cosine of 6 a thing someone could reasonably work out by hand.
I didn’t finish writing before the end of the movie, even though it took about eighteen hours to wrap up ten minutes of story. My love came home from a walk and we were talking. Anyway, this is plenty of comic strips for the week. When there are more to write about, I’ll try to have them in an essay at this link. Thanks for reading.
Comic Strip Master Command decided I should have a week to catch up on things, and maybe force me to write something original. Of all the things I read there were only four strips that had some mathematics content. And three of them are such glancing mentions that I don’t feel it proper to include the strip. So let me take care of this.
Mark Anderson’s Andertoons for the 20th is the Mark Anderson’s Andertoons for the week. Wavehead apparently wants to know whether or is the better of these equivalent forms. I understand the impulse. Rarely in real life do we see two things that are truly equivalent; there’s usually some way in which one is better than the other. There may be two ways to get home for example, both taking about the same time to travel. One might have better scenery, though, or involve fewer difficult turns or less traffic this time of day. This is different, though: or are two ways to describe the same number. Which one is “better”?
The only answer is, better for what? What do you figure to do with this number afterwards? I admit, and suppose most people have, a preference for . But that’s trained into us, in large part, by homework set to reduce fractions to “lowest terms”. There’s honest enough reasons behind that. It seems wasteful to have a factor in the numerator that’s immediately divided out by the denominator.
If this were 25 years ago, I could ask how many of you have written out a check for twenty-two and 3/4 dollars, then, rather than twenty-two and 75/100 dollars? The example is dated but the reason to prefer an equivalent form is not. If I know that I need the number represented by , and will soon be multiplying it by eight, then may save me the trouble of thinking what three times two is. Or if I’ll be adding it to , or something like that. If I’m measuring this for a recipe I need to cut in three, because the original will make three dozen cookies and I could certainly eat three dozen cookies, then may be more convenient than . What is the better depends on what will clarify the thing I want to do.
A significant running thread throughout all mathematics, not just arithmetic, is finding equivalent forms. Ways to write the same concept, but in a way that makes some other work easier. Or more likely to be done correctly. Or, if the equivalent form is more attractive, more likely to be learned or communicated. It’s of value.
Jan Eliot’s Stone Soup Classics rerun for the 20th is a joke about how one can calculate what one is interested in. In this case, going from the number of days left in school to the number of hours and minutes and even seconds left. Personally, I have never had trouble remembering there are 24 hours in the day, nor that there are 86,400 seconds in the day. That there are 1,440 minutes in the day refuses to stick in my mind. Your experiences may vary.
Harry Bliss’s Bliss for the 23rd speaks of “a truck driver with a PhD in mathematical logic”. It’s an example of signifying intelligence through mathematics credentials. (It’s also a bit classicist, treating an intelligent truck driver as an unlikely thing.)
Ted Shearer’s Quincy rerun for the 15th is one in the lineage of strips about never using mathematics in later life. Quincy challenges us to think of a time a reporter asks the President how much is 34 times 587.
That’s an unpleasant multiplication to do. But I can figure some angles on it. 34 is just a bit over one-third of 100. 587 is just a bit under 600. So, 34 times 587 has to be tolerably near one-third of 100 times 600. So it should be something around 20,000. To get it more exact: 587 is 13 less than 600. So, 587 times one-third of a hundred will be 600 times one-third of a hundred minus 13 times one-third of a hundred. That’s one-third of 130, which is about 40. So the product has to be something close to 19,960. And the product has be some number which ends in an 8, what with 4 times 7 being 28. So the answer has to be one of 19,948, 19,958, or 19,968. And, indeed, it’s 19,958. I doubt I could do that so well during a press conference, I’ll admit. (If I wanted to be sure about that second digit, I’d have worked out: the tens unit in 34 times the ones in 587 is three times seven which is 21; the ones unit in 34 times the tens unit in 587 is four times eight which is 32; and the 4 times 7 being 28 gives me a 2 in the tens unit. So, 1 plus 2 plus 2 is 5, and there we go.)
Brian Anderson’s Dog Eat Doug for the 15th uses blackboards full of equations to represent deep thinking. I can’t make out what the symbols say. They look quite good, though, and seem to have the form of legitimate expressions.
Terri Liebenson’s The Pajama Diaries for the 17th imagines creating a model for the volume of a laundry pile. The problem may seem trivial, but it reflects an important kind of work. Many processes are about how something that’s always accumulating will be handled. There’s usually a hard limit to the rate at which whatever it is gets handled. And there’s usually very little reserve, in either capacity or time. This will cause, for example, a small increase in traffic in a neighborhood to produce great jams, or how a modest rain can overflow the whole city’s sewer systems. Or how a day of missing the laundry causes there to be a week’s backlog of dirty clothes.
And a little final extra comic strip. I don’t generally mention web comics here, except for those that have fallen in with a syndicator like GoComics.com. (This is not a value judgement against web comics. It’s that I have to stop reading sometime.) But Kat Swenski’s KatRaccoon Comics recently posted this nice sequence with a cat facing her worst fear: a calculus date.
Now at last I turn to last week’s mathematically-themed comic strips. They weren’t very deeply mathematical, I think. But I always think that right before I turn out a 2,000-word essay about some kid giving a snarky answer to an arithmetic problem.
Rick Detorie’s One Big Happy for the 12th has Ruthie try to teach her brother about number words. What Ruthie seems to be struggling with is the difference between a number and the name we give a number. The distinction between a thing and the name of a thing can be a tricky one, and I do remember being confused at the difference between the word “four” and the concept “four”. What I don’t remember, to my regret, is what thought I had which made the difference clear.
Dana Simpson’s Ozy and Millie rerun for the 9th is part of a sequence of Ozy being home-schooled. The joke puts the transient nature of knowledge up against the apparent permanent of arithmetic. The joke does get at one of those fundamental questions in the philosophy of mathematics: is mathematics created or discovered? The expression of mathematics is unmistakably created. There is nothing universal in declaring “six times eight is forty-eight” and if you wish to say there is, then ask someone who speaks only Tamil and not a word of English whether they agree with exactly that proposition.
But, grant that while we may have different representations of the concept, it is the case that “eight” exists, right? We get right back into trouble if we follow up by asking, all right, will “eight” fit in my hand? Is “eight” larger than the weather? Is “eight” more or less red than nominalism? I chose nouns that made those questions obviously ridiculous. But if we want to talk about a mathematical construct existing, someone’s going to ask what traits that existence implies. It’s convenient for mathematicians, and good publicity, for us to think that we work on things that exist independently of the accidental facts of the universe. But then we’re stuck when we’re asked how we, stuck in the universe, can have anything to do with a thing that’s not part of it.
Not mentioned in this particular Ozy and Millie strip is that the characters are Buddhist. The (American) pop culture interpretation of Buddhism includes an emphasis on understanding the transient nature of … everything … which would seem to include mathematical knowledge. Still, there is a long history of great mathematical work done by Buddhist scholars; the oldest known manuscript of Indian mathematics is written in a Buddhist Hybrid Sanskrit. The author of that manuscript is unknown, but it’s not as if that were the lone piece of mathematical writing.
My limited understanding is that Indian mathematics used an interesting twist on the problem of the excluded middle. This is a question important to proofs. Can we take every logical proposition as being either true or false? If we can, then we are able to prove statements by contradiction: suppose the reverse of what we want to prove and show that implies nonsense. This is common in western mathematics. But there is a school of thought that we should not do this, and only allow as true statements we have directly proven to be true. My understanding is that at least one school of Indian mathematics allowed proof by contradiction if it proved that a thing did not exist. It would not be used to show that a thing existed. So, for example, it would allow the ordinary proof that the square root of two can’t be a rational number; it would not allow an indirect proof that, say, a kind of mapping must have a fixed point. (It would allow a proof that showed you how to find that point, though.) It’s an interesting division, and a reminder that even what counts as a logical derivation is a matter of custom.
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.
As much as everything is still happening, and so much, there’s still comic strips. I’m fortunately able here to focus just on the comics that discuss some mathematical theme, so let’s get started in exploring last week’s reading. Worth deeper discussion are the comics that turn up here all the time.
Lincoln Peirce’s Big Nate for the 5th is a casual mention. Nate wants to get out of having to do his mathematics homework. This really could be any subject as long as it fit the word balloon.
Not much to talk about there. But there is a fascinating thing about perimeters that you learn if you go far enough in Calculus. You have to get into multivariable calculus, something where you integrate a function that has at least two independent variables. When you do this, you can find the integral evaluated over a curve. If it’s a closed curve, something that loops around back to itself, then you can do something magic. Integrating the correct function on the curve around a shape will tell you the enclosed area.
And this is an example of one of the amazing things in multivariable calculus. It tells us that integrals over a boundary can tell us something about the integral within a volume, and vice-versa. It can be worth figuring out whether your integral is better solved by looking at the boundaries or at the interiors.
Heron’s Formula, for the area of a triangle based on the lengths of its sides, is an expression of this calculation. I don’t know of a formula exactly like that for the perimeter of a quadrilateral, but there are similar formulas if you know the lengths of the sides and of the diagonals.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 5th depicts, fairly, the sorts of things that excite mathematicians. The number discussed here is about algorithmic complexity. This is the study of how long it takes to do an algorithm. How long always depends on how big a problem you are working on; to sort four items takes less time than sorting four million items. Of interest here is how much the time to do work grows with the size of whatever you’re working on.
The mathematician’s particular example, and I thank dtpimentel in the comments for finding this, is about the Coppersmith–Winograd algorithm. This is a scheme for doing matrix multiplication, a particular kind of multiplication and addition of squares of numbers. The squares have some number N rows and N columns. It’s thought that there exists some way to do matrix multiplication in the order of N2 time, that is, if it takes 10 time units to multiply matrices of three rows and three columns together, we should expect it takes 40 time units to multiply matrices of six rows and six columns together. The matrix multiplication you learn in linear algebra takes on the order of N3 time, so, it would take like 80 time units.
We don’t know the way to do that. The Coppersmith–Winograd algorithm was thought, after Virginia Vassilevska Williams’s work in 2011, to take something like N2.3728642 steps. So that six-rows-six-columns multiplication would take slightly over 51.796 844 time units. In 2014, François le Gall found it was no worse than N2.3728639 steps, so this would take slightly over 51.796 833 time units. The improvement doesn’t seem like much, but on tiny problems it never does. On big problems, the improvement’s worth it. And, sometimes, you make a good chunk of progress at once.
This little essay should let me wrap up the rest of the comic strips from the past week. Most of them were casual mentions. At least I thought they were when I gathered them. But let’s see what happens when I actually write my paragraphs about them.
Thaves’s Frank and Ernest for the 2nd is a bit of wordplay, having Euclid and Galileo talking about parallel universes. I’m not sure that Galileo is the best fit for this, but I’m also not sure there’s another person connected who could be named. It’d have to be a name familiar to an average reader as having something to do with geometry. Pythagoras would seem obvious, but the joke is stronger if it’s two people who definitely did not live at the same time. Did Euclid and Pythagoras live at the same time? I am a mathematics Ph.D. and have been doing pop mathematics blogging for nearly a decade now, and I have not once considered the question until right now. Let me look it up.
It doesn’t make any difference. The comic strip has to read quickly. It might be better grounded to post Euclid meeting Gauss or Lobachevsky or Euler (although the similarity in names would be confusing) but being understood is better than being precise.
Stephan Pastis’s Pearls Before Swine for the 2nd is a strip about the foolhardiness of playing the lottery. And it is foolish to think that even a $100 purchase of lottery tickets will get one a win. But it is possible to buy enough lottery tickets as to assure a win, even if it is maybe shared with someone else. It’s neat that an action can be foolish if done in a small quantity, but sensible if done in enough bulk.
Mark Anderson’s Andertoons for the 3rd is the Mark Anderson’s Andertoons for the week. Wavehead has made a bunch of failed attempts at subtracting seven from ten, but claims it’s at least progress that some thing have been ruled out. I’ll go along with him that there is some good in ruling out wrong answers. The tricky part is in how you rule them out. For example, obvious to my eye is that the correct answer can’t be more than ten; the problem is 10 minus a positive number. And it can’t be less than zero; it’s ten minus a number less than ten. It’s got to be a whole number. If I’m feeling confident about five and five making ten, then I’d rule out any answer that isn’t between 1 and 4 right away. I’ve got the answer down to four guesses and all I’ve really needed to know is that 7 is greater than five but less than ten. That it’s an even number minus an odd means the result has to be odd; so, it’s either one or three. Knowing that the next whole number higher than 7 is an 8 says that we can rule out 1 as the answer. So there’s the answer, done wholly by thinking of what we can rule out. Of course, knowing what to rule out takes some experience.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th is the Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th for the week. It shows in joking but not wrong fashion a mathematical physicist’s encounters with orbital mechanics. Orbital mechanics are a great first physics problem. It’s obvious what they’re about, and why they might be interesting. And the mathematics of it is challenging in ways that masses on springs or balls shot from cannons aren’t.
A few problems are very easy, like, one thing in circular orbit of another. A few problems are not bad, like, one thing in an elliptical or hyperbolic orbit of another. All our good luck runs out once we suppose the universe has three things in it. You’re left with problems that are doable if you suppose that one of the things moving is so tiny that it barely exists. This is near enough true for, for example, a satellite orbiting a planet. Or by supposing that we have a series of two-thing problems. Which is again near enough true for, for example, a satellite travelling from one planet to another. But these is all work that finds approximate solutions, often after considerable hard work. It feels like much more labor to smaller reward than we get for masses on springs or balls shot from cannons. Walking off to a presumably easier field is understandable. Unfortunately, none of the other fields is actually easier.
Pythagoras died somewhere around 495 BC. Euclid was born sometime around 325 BC. That’s 170 years apart. So Pythagoras was as far in Euclid’s past as, oh, Maria Gaetana Agnesi is to mine.
I think few will oppose me if I say the best part of March 2020 was that it ended. Let me close out nearly all my March business by getting through the last couple comic strips which mentioned some mathematics topic that month. I’ll still have my readership review, probably to post Friday, and then that finishes my participation in the month at last.
Connie Sun’s Connie to the for the 30th features the title character trying to explain what “exponential growth” is. She struggles. Appropriately, as it’s something we see very rarely in ordinary life.
They turn up in mathematics all the time. And mathematical physics, and such. Any process with a rate of change that’s proportional to the current amount of the thing tends to be exponential. This whether growing or decaying. Even circular motion, periodic motion, can be understood as exponential growth with imaginary numbers. So anyone doing mathematics gets trained to see, and expect, exponentials. They have great analytic properties, too. You can use them to solve differential equations. And differential equations are so much of science that it’s easy to forget they’re not.
In ordinary life, though? Well, yes, a lot of quantities will change at rates which depend on their current quantity. But in anything that’s been around a while, the quantity will usually be at, or near enough, an equilibrium. Some kind of balance. It may move away from that balance, but usually, it’ll move back towards it. (I am skipping some complicating factors. Don’t worry about them.) A mathematician will see the hidden exponentials in this. But to anyone else? The thing may start growing, but then it peters out and slows to a stop. Or it might collapse, but that change also peters out. Maybe it’ll hit a new equilibrium; maybe it’ll go back to the old. We rarely see something changing without the sorts of limits that tamp the change back down.
Even the growth of infection rates for Covid-19 will not stay exponential forever, even if there were no public health measures responding to it. There can’t be more people infected than there are people in the world. At some point, the curve representing number of infected people versus time would stop growing more and more, and would level out, from a pattern called the logistic equation. But the early stages of this are almost indistinguishable from exponential growth.
Todd Clark’s Lola for the 30th has a student asking what the end of mathematics is. And learning how after algebra comes geometry, trigonometry, calculus, topology, and more. All fair enough, though I’m surprised to see it put for that that of course someone who does enough mathematics will do topology. (I only have a casual brush with it myself, mostly in service to other topics.) But it’s nice to have it acknowledged that, if you want, you can go on learning new mathematics fields, practically without limit.
Ashleigh Brilliant’s Pot-Shots for the 30th just declares infinity to be a favorite number. Is it a number? … We have to be careful what exactly we mean by number. Allow that we are careful, though. It’s certainly at least number-adjacent.
I know; I’m more than a week behind the original publication of these strips. The Playful Math Education Blog Carnival took a lot of what attention I have these days. I’ll get caught up again soon enough. Comic Strip Master Command tried to help me, by having the close of a week ago being pretty small mathematics mentions, too. For example:
Craig Boldman and Henry Scarpelli’s Archie for the 27th has Moose struggling in mathematics this term. This is an interesting casual mention; the joke, of Moose using three words to describe a thing he said he could in two, would not fit sharply for anything but mathematics. Or, possibly, a measuring class, but there’s no high school even in fiction that has a class in measuring.
Bud Blake’s Vintage Tiger for the 27th has Tiger and Hugo struggling to find adjective forms for numbers. We can giggle at Hugo struggling for “quadruple” and going for something that makes more sense. We all top out somewhere, though, probably around quintuple or sextuple. I have never known anyone who claimed to know what the word would be for anything past decuple, and even staring at the dictionary page for “decuple” I don’t feel confident in it.
Hilary Price’s Rhymes With Orange for the 28th uses a blackboard full of calculations as shorthand for real insight into science. From context they’re likely working on some physics problem and it’s quite hard to do that without mathematics, must agree.
John Deering’s Strange Brew for the 28th name-drops slide rules, which, yeah, have mostly historical or symbolic importance these days. There might be some niche where they’re particularly useful (besides teaching logarithms), but I don’t know of it.