Reading the Comics, February 10, 2018: I Meant To Post This Thursday Edition

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 and reprinted the 9th of February, 2018. I’m not sure just what mathematics homework Quincy could be doing to inspire him to write a book, but then, it’s not like my mind doesn’t drift while doing mathematics either. And book-writing’s a common enough daydream that most people are too sensible to act on.

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 and reprinted the 10th of February, 2018. I realized I didn’t know the father’s name and looked it up, and Wikipedia revealed to me that he’s named Redeye. You know, like the comic strip implies right there in the title. Look, I just read the comics, I can’t be expected to think about the comics too.

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.

Reading the Comics, October 21, 2017: Education Week Edition

Comic Strip Master Command had a slow week for everyone. This is odd since I’d expect six to eight weeks ago, when the comics were (probably) on deadline, most (United States) school districts were just getting back to work. So education-related mathematics topics should’ve seemed fresh. I think I can make that fit. No way can I split this pile of comics over two days.

Hector D Cantu and Carlos Castellanos’s Baldo for the 17th has Gracie quizzed about percentages of small prices, apparently as a test of her arithmetic. Her aunt has other ideas in mind. It’s hard to dispute that this is mathematics people use in real life. The commenters on GoComics got into an argument about whether Gracie gave the right answers, though. That is, not that 20 percent of $5.95 is anything about $1.19. But did Tia Carmen want to know what 20 percent of $5.95, or did she want to know what $5.95 minus 20 percent of that price was? Should Gracie have answered $4.76 instead? It took me a bit to understand what the ambiguity was, but now that I see it, I’m glad I didn’t write a multiple-choice test with both $1.19 and $4.76 as answers. I’m not sure how to word the questions to avoid ambiguity yet still sound like something one of the hew-mons might say.

Dan Thompson’s Brevity for the 19th uses the blackboard and symbols on it as how a mathematician would prove something. In this case, love. Arithmetic’s a good visual way of communicating the mathematician at work here. I don’t think a mathematician would try arguing this in arithmetic, though. I mean if we take the premise at face value. I’d expect an argument in statistics, so, a mathematician showing various measures of … feelings or something. And tests to see whether it’s plausible this cluster of readings could come out by some reason other than love. If that weren’t used, I’d expect an argument in propositional logic. And that would have long strings of symbols at work, but they wouldn’t look like arithmetic. They look more like Ancient High Martian. Just saying.

Reza Farazmand’s Poorly Drawn Lines for the 20th you maybe already saw going around your social media. It’s well-designed for that. Also for grad students’ office doors.

Dave Coverly’s Speed Bump for the 20th is designed with crossover appeal in mind and I wonder if whoever does Reading the Comics for English Teacher Jokes is running this same strip in their collection for the week.

Darrin Bell’s Candorville for the 21st sees Lemont worry that he’s forgotten how to do long division. And, fair enough: any skill you don’t use in long enough becomes stale, whether it’s division or not. You have to keep in practice and, in time, have to decide what you want to keep in practice about. (That said, I have a minor phobia about forgetting how to prove the Contraction Mapping Theorem, as several professors in grad school stressed how it must always be possible to give a coherent proof of that, even if you’re startled awake in the middle of the night by your professor.) Me, I would begin by estimating what 4,858.8 divided by 297.492 should be. 297.492 is very near 300. And 4,858.8 is a little over 4800. And that’s suggestive because it’s obvious that 48 divided by 3 is 16. Well, it’s obvious to me. So I would expect the answer to be “a little more than 16” and, indeed, it’s about 16.3.

(Don’t read the comments on GoComics. There’s some slide-rule-snobbishness, and some snark about the uselessness of the skill or the dumbness of Facebook readers, and one comment about too many people knowing how to multiply by someone who’s reading bad population-bomb science fiction of the 70s.)

Reading the Comics, September 24, 2017: September 24, 2017 Edition

Comic Strip Master Command sent a nice little flood of comics this week, probably to make sure that I transitioned from the A To Z project to normal activity without feeling too lost. I’m going to cut the strips not quite in half because I’m always delighted when I can make a post that’s just a single day’s mathematically-themed comics. Last Sunday, the 24th of September, was such a busy day. I’m cheating a little on what counts as noteworthy enough to talk about here. But people like comic strips, and good on them for liking them.

Norm Feuti’s Gil for the 24th sees Gil discover and try to apply some higher mathematics. There’s probably a good discussion about what we mean by division to explain why Gil’s experiment didn’t pan out. I would pin it down to eliding the difference between “dividing in half” and “dividing by a half”, which is a hard one. Terms that seem almost alike but mean such different things are probably the hardest part of mathematics.

Norm Feuti’s Gil for the 24th of September, 2017, didn’t appear on Gocomics.com or Comics Kingdom, my usual haunts for these comics. But I started reading the strip when it was on Comics Kingdom, and keep reading its reruns. Feuti has continued the comic strip on his own web site, and posts it on Twitter. So it’s quite easy to pick the strip back up, if you have a Twitter account or can read RSS from it. I assume you can read RSS from it.

Russell Myers’s Broom Hilda looks like my padding. But the last panel of the middle row gets my eye. The squirrels talk about how on the equinox night and day “can never be of identical length, due to the angular size of the sun and atmospheric refraction”. This is true enough for the equinox. While any spot on the Earth might see twelve hours facing the sun and twelve hours facing away, the fact the sun isn’t a point, and that the atmosphere carries light around to the “dark” side of the planet, means daylight lasts a little longer than night.

Ah, but. This gets my mathematical modelling interest going. Because it is true that, at least away from the equator, there’s times of year that day is way shorter than night. And there’s times of year that day is way longer than night. Shouldn’t there be some time in the middle when day is exactly equal to night?

The easy argument for is built on the Intermediate Value Theorem. Let me define a function, with domain each of the days of the year. The range is real numbers. It’s defined to be the length of day minus the length of night. Let me say it’s in minutes, but it doesn’t change things if you argue that it’s seconds, or milliseconds, or hours, if you keep parts of hours in also. So, like, 12.015 hours or something. At the height of winter, this function is definitely negative; night is longer than day. At the height of summer, this function is definitely positive; night is shorter than day. So therefore there must be some time, between the height of winter and the height of summer, when the function is zero. And therefore there must be some day, even if it isn’t the equinox, when night and day are the same length

There’s a flaw here and I leave that to classroom discussions to work out. I’m also surprised to learn that my onetime colleague Dr Helmer Aslaksen’s grand page of mathematical astronomy and calendar essays doesn’t seem to have anything about length of day calculations. But go read that anyway; you’re sure to find something fascinating.

Mike Baldwin’s Cornered features an old-fashioned adding machine being used to drown an audience in calculations. Which makes for a curious pairing with …

Bill Amend’s FoxTrot, and its representation of “math hipsters”. I hate to encourage Jason or Marcus in being deliberately difficult. But there are arguments to make for avoiding digital calculators in favor of old-fashioned — let’s call them analog — calculators. One is that people understand tactile operations better, or at least sooner, than they do digital ones. The slide rule changes multiplication and division into combining or removing lengths of things, and we probably have an instinctive understanding of lengths. So this should train people into anticipating what a result is likely to be. This encourages sanity checks, verifying that an answer could plausibly be right. And since a calculation takes effort, it encourages people to think out how to arrange the calculation to require less work. This should make it less vulnerable to accidents.

I suspect that many of these benefits are what you get in the ideal case, though. Slide rules, and abacuses, are no less vulnerable to accidents than anything else is. And if you are skilled enough with the abacus you have no trouble multiplying 18 by 7, you probably would not find multiplying 17 by 8 any harder, and wouldn’t notice if you mistook one for the other.

Jef Mallett’s Frazz asserts that numbers are cool but the real insight is comparisons. And we can argue that comparisons are more basic than numbers. We can talk about one thing being bigger than another even if we don’t have a precise idea of numbers, or how to measure them. See every mathematics blog introducing the idea of different sizes of infinity.

Bill Whitehead’s Free Range features Albert Einstein, universal symbol for really deep thinking about mathematics and physics and stuff. And even a blackboard full of equations for the title panel. I’m not sure whether the joke is a simple absent-minded-professor joke, or whether it’s a relabelled joke about Werner Heisenberg. Absent-minded-professor jokes are not mathematical enough for me, so let me point once again to American Cornball. They’re the first subject in Christopher Miller’s encyclopedia of comic topics. So I’ll carry on as if the Werner Heisenberg joke were the one meant.

Heisenberg is famous, outside World War II history, for the Uncertainty Principle. This is one of the core parts of quantum mechanics, under which there’s a limit to how precisely one can know both the position and momentum of a thing. To identify, with absolutely zero error, where something is requires losing all information about what its momentum might be, and vice-versa. You see the application of this to a traffic cop’s question about knowing how fast someone was going. This makes some neat mathematics because all the information about something is bundled up in a quantity called the Psi function. To make a measurement is to modify the Psi function by having an “operator” work on it. An operator is what we call a function that has domains and ranges of other functions. To measure both position and momentum is equivalent to working on Psi with one operator and then another. But these operators don’t commute. You get different results in measuring momentum and then position than you do measuring position and then momentum. And so we can’t know both of these with infinite precision.

There are pairs of operators that do commute. They’re not necessarily ones we care about, though. Like, the total energy commutes with the square of the angular momentum. So, you know, if you need to measure with infinite precision the energy and the angular momentum of something you can do it. If you had measuring tools that were perfect. You don’t, but you could imagine having them, and in that case, good. Underlying physics wouldn’t spoil your work.

Probably the panel was an absent-minded professor joke.

Reading the Comics, September 9, 2017: First Split Week Edition, Part 2

I don’t actually like it when a split week has so many more comics one day than the next, but I also don’t like splitting across a day if I can avoid it. This week, I had to do a little of both since there were so many comic strips that were relevant enough on the 8th. But they were dominated by the idea of going back to school, yet.

Randy Glasbergen’s Glasbergen Cartoons rerun for the 8th is another back-to-school gag. And it uses arithmetic as the mathematics at its most basic. Arithmetic might not be the most fundamental mathematics, but it does seem to be one of the parts we understand first. It’s probably last to be forgotten even on a long summer break.

Mark Pett’s Mr Lowe rerun for the 8th is built on the familiar old question of why learn arithmetic when there’s computers. Quentin is unconvinced of this as motive for learning long division. I’ll grant the case could be made better. I admit I’m not sure how, though. I think long division is good as a way to teach, especially, the process of estimating and improving estimates of a calculation. There’s a lot of real mathematics in doing that.

Guy Gilchrist’s Nancy for the 8th is another back-to-school strip. Nancy’s faced with “this much math” so close to summer. Her given problem’s a bit of a mess to me. But it’s mostly teaching whether the student’s got the hang of the order of operations. And the instructor clearly hasn’t got the sense right. People can ask whether we should parse “12 divided by 3 times 4” as “(12 divided by 3) times 4” or as “12 divided by (3 times 4)”, and that does make a major difference. Multiplication commutes; you can do it in any order. Division doesn’t. Leaving ambiguous phrasing is the sort of thing you learn, instinctively, to avoid. Nancy would be justified in refusing to do the problem on the grounds that there is no unambiguous way to evaluate it, and that the instructor surely did not mean for her to evaluate it all four different plausible ways.

By the way, I’ve seen going around Normal Person Twitter this week a comment about how they just discovered the division symbol ÷, the obelus, is “just” the fraction bar with dots above and below where the unknown numbers go. I agree this is a great mnemonic for understanding what is being asked for with the symbol. But I see no evidence that this is where the symbol, historically, comes from. We first see ÷ used for division in the writings of Johann Henrich Rahn, in 1659, and the symbol gained popularity particularly when John Pell picked it up nine years later. But it’s not like Rahn invented the symbol out of nowhere; it had been used for subtraction for over 125 years at that point. There were also a good number of writers using : or / or \ for division. There were some people using a center dot before and after a / mark for this, like the % sign fell on its side. That ÷ gained popularity in English and American writing seems to be a quirk of fate, possibly augmented by it being relatively easy to produce on a standard typewriter. (Florian Cajori notes that the National Committee on Mathematical Requirements recommended dropping ÷ altogether in favor of a symbol that actually has use in non-mathematical life, the / mark. The Committee recommended this in 1923, so you see how well the form agenda is doing.)

Dave Whamond’s Reality Check for the 8th is the anthropomorphic-numerals joke for this week. A week without one is always a bit … peculiar.

Mark Leiknes’s Cow and Boy rerun for the 9th only mentions mathematics, and that as a course that Billy would rather be skipping. But I like the comic strip and want to promote its memory as much as possible. It’s a deeply weird thing, because it has something like 400 running jokes, and it’s hard to get into because the first couple times you see a pastoral conversation interrupted by an orca firing a bazooka at a cat-helicopter while a panda brags of blowing up the moon it seems like pure gibberish. If you can get through that, you realize why this is funny.

Dave Blazek’s Loose Parts for the 9th uses chalkboards full of stuff as the sign of a professor doing serious thinking. Mathematics is will-suited for chalkboards, at least in comic strips. It conveys a lot of thought and doesn’t need much preplanning. Although a joke about the difficulties in planning out blackboard use does take that planning. Yes, there is a particular pain that comes from having more stuff to write down in the quick yet easily collaborative medium of the chalkboard than there is board space to write.

Brian Basset’s Red and Rover for the 9th also really only casually mentions mathematics. But it’s another comic strip I like a good deal so would like to talk up. Anyway, it does show Red discovering he doesn’t mind doing mathematics when he sees the use.

Reading the Comics, April 5, 2016: April 5, 2016 Edition

I’ve mentioned I like to have five or six comic strips for a Reading The Comics entry. On the 5th, it happens, I got a set of five all at once. Perhaps some are marginal for mathematics content but since when does that stop me? Especially when there’s the fun of a single-day Reading The Comics post to consider. So here goes:

Mark Anderson’s Andertoons is a student-resisting-the-problem joke. And it’s about long division. I can’t blame the student for resisting. Long division’s hard to learn. It’s probably the first bit of arithmetic in which you just have to make an educated guess for an answer and face possibly being wrong. And this is a problem that’ll have a remainder in it. I think I remember early on in long division finding a remainder left over feeling like an accusation. Surely if I’d done it right, the divisor would go into the original number a whole number of times, right? No, but you have to warm up to being comfortable with that.

Ted Key’s Hazel rerun the 5th of April, 2016. Were the Pac-Man ghosts ever called space demons? It seems like they might’ve been described that way in some boring official manual that nobody ever read. I always heard them as “ghosts” anyway.

Ted Key’s Hazel feels less charmingly out-of-date when you remember these are reruns. Ted Key — who created Peabody’s Improbable History as well as the sitcom based on this comic panel — retired in 1993. So Hazel’s attempt to create a less abstract version of the mathematics problem for Harold is probably relatively time-appropriate. And recasting a problem as something less abstract is often a good way to find a solution. It’s all right to do side work as a way to get the work you want to do.

John McNamee’s Pie Comic is a joke about the uselessness of mathematics. Tch. I wonder if the problem here isn’t the abstractness of a word like “hypotenuse”. I grant the word doesn’t evoke anything besides “hypotenuse”. But one irony is that hypotenuses are extremely useful things. We can use them to calculate how far away things are, without the trouble of going out to the spot. We can imagine post-apocalyptic warlords wanting to know how far things are, so as to better aim the trebuchets.

Percy Crosby’s Skippy is a rerun from 1928, of course. It’s also only marginally on point here. The mention of arithmetic is irrelevant to the joke. But it’s a fine joke and I wanted people to read it. Longtime readers know I’m a Skippy fan. (Saturday’s strip follows up on this. It’s worth reading too.)

Bill Griffith’s Zippy the Pinhead for the 5th of April, 2016. Not what Neil DeGrasse Tyson is like when he’s not been getting enough sleep.

Bill Griffith’s Zippy the Pinhead has picked up some quantum mechanics talk. At least he’s throwing around the sorts of things we see in pop science and, er, pop mathematical talk about the mathematics of cutting-edge physics. I’m not aware of any current models of everything which suppose there to be fourteen, or seventeen, dimensions of space. But high-dimension spaces are common points of speculation. Most of those dimensions appear to be arranged in ways we don’t see in the everyday world, but which leave behind mathematical traces. The crack about God not playing dice with the universe is famously attributed to Albert Einstein. Einstein was not comfortable with the non-deterministic nature of quantum mechanics, that there is this essential randomness to this model of the world.

Reading the Comics, December 27, 2014: Last of the Year Edition?

I’m curious whether this is going to be the final bunch of mathematics-themed comics for the year 2014. Given the feast-or-famine nature of the strips it’s plausible we might not have anything good through to mid-January, but, who knows? Of the comics in this set I think the first Peanuts the most interesting to me, since it’s funny and gets at something big and important, although the Ollie and Quentin is a better laugh.

Mark Leiknes’s Cow and Boy (December 23, rerun) talks about chaos theory, the notion that incredibly small differences in a state can produce enormous differences in a system’s behavior. Chaos theory became a pop-cultural thing in the 1980s, when Edward Lorentz’s work (of twenty years earlier) broke out into public consciousness. In chaos theory the chaos isn’t that the system is unpredictable — if you have perfect knowledge of the system, and the rules by which it interacts, you could make perfect predictions of its future. What matters is that, in non-chaotic systems, a small error will grow only slightly: if you predict the path of a thrown ball, and you have the ball’s mass slightly wrong, you’ll make a proportionately small error on what the path is like. If you predict the orbit of a satellite around a planet, and have the satellite’s starting speed a little wrong, your prediction is proportionately wrong. But in a chaotic system there are at least some starting points where tiny errors in your understanding of the system produce huge differences between your prediction and the actual outcome. Weather looks like it’s such a system, and that’s why it’s plausible that all of us change the weather just by existing, although of course we don’t know whether we’ve made it better or worse, or for whom.

Charles Schulz’s Peanuts (December 23, rerun from December 26, 1967) features Sally trying to divide 25 by 50 and Charlie Brown insisting she can’t do it. Sally’s practical response: “You can if you push it!” I am a bit curious why Sally, who’s normally around six years old, is doing division in school (and over Christmas break), but then the kids are always being assigned Thomas Hardy’s Tess of the d’Urbervilles for a book report and that is hilariously wrong for kids their age to read, so, let’s give that a pass.

Reading the Comics, October 13, 2012

I suppose it’s been long enough to resume the review of math-themed comic strips. I admit there are weeks I don’t have much chance to write regular articles and then I feel embarrassed that I post only comic strips links, but I do enjoy the comics and the comic strip reviews. This one gets slightly truncated because King Features Syndicate has indeed locked down their Comics Kingdom archives of its strips, making it blasted inconvenient to read and nearly impossible to link to them in any practical, archivable way. They do offer a service, DailyInk.com, with their comic strips, but I can hardly expect every reader of mine to pay up over there just for the odd day when Mandrake the Magician mentions something I can build a math problem from. Until I work out an acceptable-to-me resolution, then, I’ll be dropping to gocomics.com and a few oddball strips that the Houston Chronicle carries.