Reading the Comics, June 20, 2019: Old Friends Edition


We continue to be in the summer vacation doldrums for mathematically-themed comic strips. But there’ve been a couple coming out. I could break this week’s crop into two essays, for example. All of today’s strips are comics that turn up in my essays a lot. It’s like hanging out with a couple of old friends.

Samson’s Dark Side of the Horse for the 17th uses the motif of arithmetic expressions as “difficult” things. The expressions Samson quotes seem difficult for being syntactically weird: What does the colon under the radical sign mean in \sqrt{9:}33 ? Or they’re difficult for being indirect, using a phrase like “50%” for “half”. But with some charity we can read this as Horace talking about 3:33 am to about 6:30 am. I agree that those are difficult hours.

Horace: 'I've lived through some difficult times. Especially from sqrt{9:}33 AM to 50% past sixish o'clock. Maybe I should get my watch fixed.'
Samson’s Dark Side of the Horse for the 17th of June, 2019. Some of the many essays inspired by Dark Side of the Horse are at this link.

It also puts me in mind of a gift from a few years back. An aunt sent me an Irrational Watch, with a dial that didn’t have the usual counting numbers on it. Instead there were various irrational numbers, like the Golden Ratio or the square root of 50 or the like. Also the Euler-Mascheroni Constant, a number that may or may not be irrational. Nobody knows. It’s likely that it is irrational, but it’s not proven. It’s a good bit of fun, although it does make it a bit harder to use the watch for problems like “how long is it until 4:15?” This isn’t quite what’s going on here — the square root of nine is a noticeably rational number — but it seems in that same spirit.

Mark Anderson’s Andertoons for the 18th sees Wavehead react to the terminology of the “improper fraction”. “Proper” and “improper” as words carry a suggestion of … well, decency. Like there’s something faintly immoral about having an improper fraction. “Proper” and “improper”, as words, attach to many mathematical concepts. Several years ago I wrote that “proper” amounted to “it isn’t boring”. This is a fair way to characterize, like, proper subsets or proper factors or the like. It’s less obvious that \frac{13}{12} is a boring fraction.

The teacher has on the blackboard 1/3 + 3/4 rewritten as 4/12 + 9/12 = 13/12. Wavehead: 'OK, we made it so they had something in common, added them together, and the result is *improper*? I mean, I kinda feel like we just made things worse!'
Mark Anderson’s Andertoons for the 18th of June, 2019. Essays with some mention of a topic from Andertoons are at this link.

I may need to rewrite that old essay. An “improper” form satisfies all the required conditions for the term. But it misses some of the connotation of the term. It’s true that, say, the new process takes “a fraction of the time” of the old, if the old process took one hour and the new process takes fourteen years. But if you tried telling someone that they would assume you misunderstood something. The ordinary English usage of “fraction” carries the connotation of “a fraction between zero and one”, and that’s what makes a “proper fraction”.

In practical terms, improper fractions are fine. I don’t know of any mathematicians who seriously object to them, or avoid using them. The hedging word “seriously” is in there because of a special need. That need is: how big is, say, \frac{75}{14} ? Is it bigger than five? Is it smaller than six? An improper fraction depends on you knowing, in this case, your fourteen-times tables to tell. Switching that to a mixed fraction, 5 + \frac{5}{14} , helps figure out what the number means. That’s as far as we have to worry about the propriety of fractions.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th uses the form of a Fermi problem for its joke. Fermi problems have a place in mathematical modeling. The idea is to find an estimate for some quantity. We often want to do this. The trick is to build a simple model, and to calculate using a tiny bit of data. The Fermi problem that has someone reached public consciousness is called the Fermi paradox. The question that paradox addresses is, how many technologically advanced species are there in the galaxy? There’s no way to guess. But we can make models and those give us topics to investigate to better understand the problem. (The paradox is that reasonable guesses about the model suggest there should be so many aliens that they’d be a menace to air traffic. Or that the universe should be empty except for us. Both alternatives seem unrealistic.) Such estimates can be quite wrong, of course. I remember a Robert Heinlein essay in which he explained the Soviets were lying about the size of Moscow, his evidence being he didn’t see the ship traffic he expected when he toured the city. I do not remember that he analyzed what he might have reasoned wrong when he republished this in a collection of essays he didn’t seem to realize were funny.

HR interviewer: 'At this company we only want geniuses. So we ask puzzles and judge how well you solve them. Quick! Estimate how many employees we have!' Job applicant: 'Given other companies use empirically validated non-annoying hiring protocols and that engineers have lots of options, I'd estimate your company has exactly one employee.' Interviewer: 'Please don't leave me.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th of June, 2019. Anyone who’s been reading these for a couple weeks knows, but, Saturday Morning Breakfast Cereal features in essays at this link. Hey, every essay is somebody’s first.

So the interview question presented is such a Fermi problem. The job applicant, presumably, has not committed to memory the number of employees at the company. But there would be clues. Does the company own the whole building it’s in, or just a floor? Just an office? How large is the building? How large is the parking lot? Are there people walking the hallways? How many desks are in the offices? The question could be answerable. The applicant has a pretty good chain of reasoning too.

Bill Amend’s FoxTrot Classics for the 20th has several mathematical jokes in it. One is the use of excessively many decimal points to indicate intelligence. Grant that someone cares about the hyperbolic cosines of 15.2. There is no need to cite its wrong value to nine digits past the decimal. Decimal points are hypnotic, though, and listing many of them has connotations of relentless, robotic intelligence. That is what Amend went for in the characters here. That and showing how terrible nerds are when they find some petty issue to rage over.

Eugene: 'Lousy camp-issued calculator!' Marcus: 'What's wrong now?' Eugene: 'This thing says the hyperbolic cosine of 15.2 is 0.965016494 when any moron knows this can't be right! What kin of boneheads run this palce? See? It did it again!' Marcus: 'You need to hit the blue button first. Right now you're just getting the regular cosine. ... No need to say 'thank you'. I'm enjoying this silence.' Jason: 'Did you want to borrow mine? Some of us don't need them.'
Bill Amend’s FoxTrot Classics for the 20th of June, 2019. It originally ran the 3rd of July, 1997. Essays based on FoxTrot, either the current-run Sundays, newspaper-rerun 2000s strips, or 90s-run Classics, are at this link.

Eugene is correct about the hyperbolic cosine being wrong, there, though. He’s not wrong to check that. It’s good form to have some idea what a plausible answer should be. It lets one spot errors, for one. No mathematician is too good to avoid making dumb little mistakes. And computing tools will make mistakes too. Fortunately they don’t often, but this strip originally ran a couple years after the discovery of the Pentium FDIV bug. This was a glitch in the way certain Pentium chips handled floating-point division. It was discovered by Dr Thomas Nicely, at Lynchberg College, who found inconsistencies in some calculations when he added Pentium systems to the computers he was using. This Pentium bug may have been on Amend’s mind.

Eugene would have spotted right away that the hyperbolic cosine was wrong, though, and didn’t need nine digits for it. The hyperbolic cosine is a function. Its domain is the real numbers. It range is entirely numbers greater than or equal to one, or less than or equal to minus one. A 0.9 something just can’t happen, not as the hyperbolic cosine for a real number.

And what is the hyperbolic cosine? It’s one of the hyperbolic trigonometric functions. The other trig functions — sine, tangent, arc-sine, and all that — have their shadows too. You’ll see the hyperbolic sine and hyperbolic tangent some. You will never see the hyperbolic arc-cosecant and anyone trying to tell you that you need it is putting you on. They turn up in introductory calculus classes because you can differentiate them, and integrate them, the way you can ordinary trig functions. They look just different enough from regular trig functions to seem interesting for half a class. By the time you’re doing this, your instructor needs that.

The ordinary trig functions come from the unit circle. You can relate the Cartesian coordinates of a point on the circle described by x^2 + y^2 = 1 to the angle made between that point and the center of the circle and the positive x-axis. Hyperbolic trig functions we can relate the Cartesian coordinates of a point on the hyperbola described by x^2 - y^2 = 1 to angles instead. The functions … don’t have a lot of use at the intro-to-calculus level. Again, other than that they let you do some quite testable differentiation and integration problems that don’t look exactly like regular trig functions do. They turn up again if you get far enough into mathematical physics. The hyperbolic cosine does well in describing catenaries, that is, the shape of flexible wires under gravity. And the family of functions turn up in statistical mechanics, often, in the mathematics of heat and of magnetism. But overall, these functions aren’t needed a lot. A good scientific calculator will offer them, certainly. But it’ll be harder to get them.

There is another oddity at work here. The cosine of 15.2 degrees is about 0.965, yes. But mathematicians will usually think of trigonometric functions — regular or hyperbolic — in terms of radians. This is just a different measure of angles. A right angle, 90 degrees, is measured as \frac{1}{2}\pi radians. The use of radians makes a good bit of other work easier. Mathematicians get to accustomed to using radians that to use degrees seems slightly alien. The cosine of 15.2 radians, then, would be about -0.874. Eugene has apparently left his calculator in degree mode, rather than radian mode. If he weren’t so worked up about the hyperbolic cosine being wrong he might have noticed. Perhaps that will be another exciting error to discover down the line.

This strip was part of a several-months-long story Bill Amend did, in which Jason has adventures at Math Camp. I don’t remember the whole story. But I do expect the strip to have several more appearances here this summer.


And that’s about half of last week’s comics. A fresh Reading the Comics post should be at this link later this week. Thank you for reading along.

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Reading the Comics, June 15, 2019: School Is Out? Edition


This has not been the slowest week for mathematically-themed comic strips. The slowest would be the week nothing on topic came up. But this was close. I admit this is fine as I have things disrupting my normal schedule this week. I don’t need to write too many essays too.

On-topic enough to discuss, though, were:

Lalo Alcaraz’s La Cucaracha for the 9th features a teacher trying to get ahead of student boredom. The idea that mathematics is easier to learn if it’s about problems that seem interesting is a durable one. It agrees with my intuition. I’m less sure that just doing arithmetic while surfing is that helpful. My feeling is that a problem being interesting is separate from a problem naming an intersting thing. But making every problem uniquely interesting is probably too much to expect from a teacher. A good pop-mathematics writer can be interesting about any problem. But the pop-mathematics writer has a lot of choice about what she’ll discuss. And doesn’t need to practice examples of a problem until she can feel confident her readers have learned a skill. I don’t know that there is a good answer to this.

Teacher: 'Class, today is the last day of school. You don't want to be here, and neither do I. So, I found a way where we can learn while getting an early start on the summer break!' Next panel, they're all on surfboards. Teacher: 'Next question: whats eight sick waves times eight six waves?' Students: 'Sixty-four sick waves!'
Lalo Alcaraz’s La Cucaracha for the 9th of June, 2019. I had thought I’d mentioned this comic at least a couple times in the past, and seem to be wrong. So this is a new tag and that’s always nice to have. Any future essays which mention something inspired by La Cucaracha should be at this link.

Also part of me feels that “eight sick waves times eight sick waves” has to be “sixty-four sick-waves-squared”. This is me worrying about the dimensional analysis of a joke. All right, but if it were “eight inches times eight inches” and you came back with “sixty-four inches” you’d agree something was off, right? But it’s easy to not notice the units. That we do, mechanically, the same thing in multiplying (oh) three times $1.20 or three times 120 miles or three boxes times 120 items per box as we do multiplying three times 120 encourages this. But if we are using numbers to measure things, and if we are doing calculations about things, then the units matter. They carry information about the kinds of things our calculations represent. It’s a bad idea to misuse or ignore those tools.

Paul Trap’s Thatababy for the 14th is roughly the anthropomorphized geometry cartoon of the week. It does name the three ways to group triangles based on how many sides have the same length. Or if you prefer, how many interior angles have the same measure. So it’s probably a good choice for your geometry tip sheet. “Scalene” as a word seems to have entered English in the 1730s. Its origin traces to Late Latin “scalenus”, from the Greek “skalenos” and meaning “uneven” or “crooked”.

Thatababy drawing triangles: an equilateral triangle, an isosceles triangle, a scalene triangle, and then a love triangle, showing two isosceles triangles holding hands; one of them looks with interest at an equilateral triangle.
Paul Trap’s Thatababy for the 14th of June, 2019. Now, this strip I thought I featured more around here. It doesn’t seem to have gotten an appearance in over a year, though. Still, other appearances by Thatababy should be in essays at this link.

“Isosceles” also goes to Late Latin and, before that, the Greek “isoskeles”, with “iso” the prefix meaning “equal” and “skeles” meaning “legs”. The curious thing to me is “Isosceles”, besides sounding more pleasant, came to English around 1550. Meanwhile, “equilateral” — a simple Late Latin for “equal sides” — appeared around 1570. I don’t know what was going on that it seemed urgent to have a word for triangles with two equal sides first, and a generation later triangles with three equal sides. And then triangles with no two equal sides went nearly two centuries without getting a custom term.

But, then, I’m aware of my bias. There might have been other words for these concepts, recognized by mathematicians of the year 1600, that haven’t come to us. Or it might be that scalene triangles were thought to be so boring there wasn’t any point giving them a special name. It would take deeper mathematics history knowledge than I have to say.


Those are all the mathematically-themed comic strips I can find something to discuss from the past week. There were some others with mentions of mathematics, though. These include:

Tony Rubino and Gary Markstein’s Daddy’s Home for the 9th, in which mathematics is the last class of the school year. Francesco Marciuliano and Jim Keefe’s Sally Forth for the 11th has a study session with “math charades” mentioned. Mark Andersons Andertoons for the 11th wants in on some of my sweet Thatababy exposition. Harley Schwadron’s 9 to 5 for the 14th is trying to become the default pie chart joke around here. It won’t beat out Randolph Itch, 2 am without a stronger punch line. And Mark Tatulli’s Heart of the City for the 15th sees Dean mention hiding sleeping in algebra class.


This closes out a week’s worth of comic strips. My next Reading the Comics post should be at this link next Sunday. And now I need to think of something to post for the Thursday and, if I can, Tuesday publication dates.

On The Goldfish Situation


If you’ve been following me on Twitter you’ve seen reports of the Great Migration. This is the pompous name I give to the process of bringing the goldfish who were in tanks in the basement for the winter back outside again. This to let them enjoy the benefits of the summer, like, not having me poking around testing their water every day. (We had a winter with a lot of water quality problems. I’m probably over-testing.)

My reports about moving them back — by setting a net in that could trap some fish and moving them out — included reports of how many remained in each tank. And many people told me how such updates as “Twelve goldfish are in the left tank, three in the right, and fifteen have been brought outside” sound like the start of a story problem. Maybe it does. I don’t have a particular story problem built on this. I’m happy to take nominations for such.

But I did have some mathematics essays based on the problem of moving goldfish to the pond outdoors and to the warm water tank indoors:

  • How To Count Fish, about how one could estimate a population by sampling it twice.
  • How To Re-Count Fish, about one of the practical problems in using this to count as few goldfish as we have at our household.
  • How Not To Count Fish, about how this population estimate wouldn’t work because of the peculiarities of goldfish psychology. Honest.

That I spend one essay describing how to do a thing, and then two more essays describing why it won’t work, may seem characteristically me. Well, yeah. Mathematics is a great tool. To use a tool safely requires understanding its powers and its limitations. I like thinking about what mathematics can and can’t do.

Reading the Comics, June 6, 2019: Not The Slowest Week Edition


Comic Strip Master Command started the summer vacation early this year. There have been even slower weeks for mathematically-themed comics, but not many, and not much slower. Well, it’s looking like a nice weekend anyway. We can go out and do something instead.

And I’m doing a little experiment to see what happens if I publish posts a bit earlier in the day. My suspicion is nothing that reaches statistical significance. But statistical significance isn’t everything. I can devote a month or two to a lark.

Piers Baker’s Ollie and Quentin for the 2nd is a rerun. The strip ended several years ago, and has not been one of those formerly syndicated comics gone to web-only publication. And it’s one that I’ve discussed before, in a 2014 repeat and briefly in 2015. I don’t know why it reran six months apart. Having a particular daily strip repeat so often is usually a sign I should retire the strip from this blog. Likely I won’t retire it from my reading. I like its style a bit too much.

Quentin: 'Sorry you aren't feeling happy today.' Ollie: 'Why do you think I'm not happy?' Quentin: 'Studies show 50% of people aren't happy, and I'm in a great mood.' Ollie: 'You idiot! It doesn't work like that!' Quentin: 'Yes it does, every second person isn't happy, I'm happy, so you can't be.' Ollie: 'I am happy you moron!' Quentin: 'No you're not.' Ollie: 'I AM!' Quentin: 'You don't sound it!' Ollie: 'AAAARGH!' (And he storms off, cursing.) Quentin: 'Sorry you aren't feeling happy today.'
Piers Baker’s Ollie and Quentin for the 2nd of June, 2019. I find that I’ve discussed this strip less often than I imagined. Essays including some mention of Ollie and Quentin appear at this link. There are some appearances of the strip which predate my using the comic as a tag, however.

The joke is built on Quentin hearing that only 50% of people are not happy. And as he is happy, and he and Ollie are two people, it follows Ollie can’t be. The joke builds on the logic of the gambler’s fallacy. This is the idea that the probability of some independent event depends on what has recently happened. Here “event” means what it does to statisticians, what it turns out something is. This can be the result of a coin toss. This can be finding out whether a person is happy or not. The gambler’s fallacy has a hard-to-resist logic to it. We know it is unlikely that a coin tossed fairly ten times will come up tails each time. We also know it is even more unlikely that a coin tossed fairly eleven times will turn up tails every time. So if the coin has already come up tails ten times? It’s easy in the abstract to sneer at people who make this mistake. But at some point or other we all think some unpredictable event is “due”.

There is a catch here, though. The gambler’s fallacy covers independent events. One coin’s toss does not affect whether the next toss should be heads or tails. But personal happiness? That is something affected by other people. Perhaps not dramatically. But one person’s mood can certainly alter another’s, just as the strip demonstrates. In past appearances of this strip I’ve written about it as though the mathematical comedy element were obvious. Now I realize I may have under-explored what is happening here.

Student at blackboard, working problems like 3+2 and 2+2, to the teacher: 'Do we need to learn this in case our smart devices are down?'
Harley Schwadron’s 9 to 5 for the 3rd of June, 2019. This strip I mention rarely, but that’s about as often as I expect. Essays inspired by something in 9 to 5 appear at this link.

Harley Schwadron’s 9 to 5 for the 3rd is a student-at-the-blackboard joke. And a joke about the uselessness of learning arithmetic if there are computing devices around. There have always been computing devices around, though. I’d prefer them for tedious problems, or for problems in which mistakes have serious consequences. But I think it’s worth knowing at least what to do. But I like mathematics. Of course I would.

Student at blackboard, having written out 7 x 6 = 50, to the teacher: 'I added a tip.'
Mike Baldwin’s Cornered for the 6th of June, 2019. This comic comes up sometimes. Cornered appears in essays at this link.

Mike Baldwin’s Cornered for the 6th is another student-at-the-blackboard joke. This one has the student excusing his wrong answer, a number too high, as a tip. In the student’s defense, I’ll say being able to come up with a decent approximate answer, even one you know is a little too high, is worth it. Often an important step in a problem is knowing about what a reasonable answer is. This can involve mental-mathematics tricks. For example, remembering that 7 times 7 is just under fifty, which would help with a problem like 7 times 6.


And that’s all the comic strips I found worth any mention last week. There weren’t even any that rated a “there’s a comic that said ‘math class’, so here you go” aside. This bodes well for an interesting week of content around here. My next Reading the Comics post should appear next Sunday at this link. All the past comic strip discussion should, too. If you should find a comics essay that doesn’t appear in those archives please let me know. I’ll fix it.

Reading the Comics, June 1, 2019: More Than I Thought Edition


When I collected last week’s mathematically-themed comic strips I thought this set an uninspiring one. That changed sometime while I wrote. That’s the sort of week I like to have.

Richard Thompson’s Richard’s Poor Almanac for the 28th is a repeat; all these strips are. And I’ve featured it here before too. But never before in color, so I’ll take this chance to show it one last time. One of the depicted plants is the “Non-Euclidean Creeper”, which “ignores the geometry of the space-time continuum”. Non-Euclidean is one of those few geometry-related words that people recognize — maybe even only learn — in their adulthood. It has connotations of the bizarre and the weird and the wrong.

And it is a bit weird. While we live in a non-Euclidean space, we never really notice. Euclidean space is the geometry we’re used to from drawing shapes on paper and putting boxes in the corners of basements. And from this we’ve given “non-Euclidean” this sinister reputation. We credit it with defying common sense and even logic itself, although it’s geometry. It can’t defy logic. It can defy intuition. Non-Euclidean geometries have the idea that there are no such things as parallel lines. Or the idea that there are too many parallel lines. And it can get to weird results, particularly if we look at more than three dimensions of space. Those also tax the imagination. It will get a weed a bad reputation.

Your Spring Weeding Guide. Non-Euclidean Creeper. Hard to remove. Ignores the geometry of the spacetime continuum. Common to most yard. (Picture of a woman with garden knife trying to kill a plant that grows around the other side of hte panel.) False Tea Rose. Looks and smells exactly like the lovely tea rose, but it's a weed! Soon your yard will be covered in it! Root it out! Tear it up! Kill it! (Man with rake trying to kill a bush.) Bamzu. COmbines the robust unstoppability of kudzu with the hearty immortality of bamboo. It also attracts zebra mussels. Sell your house and get a condo. (Woman trying to kill a tidal wave of plant with a rake.) Dilatory Bulbvine. Also known as your leftover Christmas lights. Take them down already, it's Easter for crying out loud. (Man saying 'whoopsie' while taking off a strand of lights.)
Richard Thompson’s Richard’s Poor Almanac for the 28th of May, 2019. And, sadly, this probably wraps up the essays I can usefully write about this strip. Essays about Richard’s Poor Almanac should be at this link.

Chen Weng’s Messycow Comics for the 30th is about a child’s delight in learning how to count. I don’t remember ever being so fascinated by counting that it would distract me permanently. I do remember thinking it was amazing that once a pattern was established it kept on, with no reason to ever stop, or even change. My recollection is I thought this somehow unfair to the alphabet, which had a very sudden sharp end.

Girl: 'Mommy, I can count to 100!' Mom: 'Show me!' Girl counts up to 98 99, 100! Mom: 'Wow! Great job! I'm so proud!' (At bedtime.) Mom: 'OK, honey, time to sleep.' Girl: '1, 2, 3, 4.' (Getting the girl off a step.) Mom: 'We are late, let's GO!' Girl: '38, 39, 50? No, 40?' (Dragging the girl out of a room on fire.) Girl '66, 67, 68, 69 ... what's next?' Mom: 'What have I done?'
Chen Weng’s Messycow Comics for the 30th of May, 2019. This is a new strip around here. This and any future essays inspired by Messycow Comics should appear at this link.

The counting numbers — counting in general — seem to be things we’ve evolved to understand. Other animals know how to count. Here I recommend again Stanislas Dehaene’s The Number Sense: How the Mind Creates Mathematics, which describes some of the things we know about how animals do mathematics. It also describes how children come to understand it.

Samson’s Dark Side of the Horse for the 31st is a bit of play with arithmetic. Horace simplifies his problem by catching all the numerals with loops in them — the zeroes and the eights — and working with what’s left. Evidently he’s already cast out all the nines. (This is me making a joke. Casting out nines is a simple checksum that you can do which can guard against some common arithmetic mistakes. It doesn’t catch everything. But it is simple enough to do that it can be worth using.)

Horace working on the problem '100 x 80008005 ='. He strikes out many of the digits from where they appear over his head. What's left is '1 x 5 =', which he answers as 5.
Samson’s Dark Side of the Horse for the 31st of May, 2019. This comic appears a lot around here. Essays including Dark Side of the Horse appear at this link.

The part that disappoints me is that to load the problem up with digits with loops, we get a problem that’s not actually hard: 100 times anything is easy. If the problem were, say, 189 times 80008005 then you’d have a problem someone might sensibly refuse to do. But without those zeroes at the start it’d be harder to understand what Horace was doing. Maybe if it were 10089 times 800805 instead.

The Hookup. At a bar, an anthropomorphic B says to an anthropomorphic 4: 'If numbers don't lie, why did your profile say you were a ten?' (Title panel gag: the 4 says, 'Try me. Let's turn B4 into after.')
Hilary Price and Rina Piccolo’s Rhymes with Orange for the 1st of June, 2019. I don’t get enough chances to write about this comic, which I like, possibly because the title panel format amuses me more than it maybe objectively should. The chances I have had to write about Rhymes With Orange are at this link.

Hilary Price and Rina Piccolo’s Rhymes with Orange for the 1st is the anthropomorphic numerals joke for the week. Also the anthropomorphic letters joke. The capital B sees occasional use in mathematics. It can represent the ball, that is, the set of all points that represent the interior of a sphere of a set radius. Usually a radius of 1. It also sometimes appears in equations as a parameter, a number whose value is fixed for the length of the problem but whose value we don’t care about. I had thought there were a few other roles for B alone, such as a label to represent the Bessel functions. These are a family of complicated-looking polynomials with some nice properties it’s too great a diversion for me to discuss just now. But they seem to more often be labelled with a capital J for reasons that probably seemed compelling at the time. It’ll also get used in logic, where B might stand for the second statement of some argument. 4, meanwhile, is that old familiar thing.


And there were a couple of comics which I like, but which mentioned mathematics so slightly that I couldn’t put a paragraph into them. Henry Scarpelli and Craig Boldman’s Archie rerun for the 27th, for example, mentions mathematics class as one it’s easy to sleep through. And Tony Cochrane’s Agnes for the 28th also mentions mathematics class, this time as one it’s hard to pay attention to.


This clears out last week’s comic strips. This present week’s strips should be at this link on Sunday. I haven’t yet read Friday or Saturday’s comics, so perhaps there’s been a flood, but this has been a slow week so far.

How May 2019 Treated My Mathematics Blog


It’s two days past when I wanted to do my self-inspection, but that’s all right. Better to have a thing done than not. I had another month of decline on the mathematics blog, inexplicable except for my going and vanishing for a week at a time without notice or much interesting content.

I published ten things in May, my quietest month in years. And the number of things I post seems to be the most important thing I can control to encourage readers. Well, I could change the time of day that I post. For several years now I’ve posted everything at 18:00 Universal Time. That’s about 2 pm Eastern Daylight Time, in my home time zone. It’s possible another hour might serve my interests in being read better.

There were 981 page views in May, down from 1,020 in April (twelve posts) and 1,391 in March (fourteen posts). It’s the first time I didn’t break a thousand since December 2017 (another eleven-post month). The number of unique visitors rose slightly, though: 721 unique visitors in May, compared to 668 in April and 954 in March. (December 2017 had 599 unique visitors.) There is probably a great deal of fluctuation in all this.

Bar chart showing about four and a half years of readership figures which are fairly constant, with a few peaks in the spring and summer of last year.
What’s fun about looking over this many months at once is trying to spot where I was running A To Z sequences. You can actually see people responding to sudden, two- and three-month stretches of regularly-published high-quality articles. It’s almost a lesson or something.

The number of likes continued to be erratic. 43 things were liked here in May, up from April’s 40, down from March’s 97. For what it’s worth the twelve-month running average leading up to May was 72 likes per month. This was an unliked month. The number of comments had one of its sporadic upticks, with 12 comments. There’d been 14 in April and a near-record-low four in March. Again for what it’s worth the twelve-month running average is 25 comments per month. That range does include some of the A To Z months, which invite comments in a way I don’t seem to be able to do normally.

163 different posts got at least one view in May. The ones that got the most were a couple perennials and one that I figured to be liked, for how many words I put into it:

The record grooves and the trapezoids people always ask about. I figured a nice meaty question like the continuity of a familiar function would get readers. What’s always a bit of a surprise is which Reading the Comics post gets the most readers in a month. Generically I’d expect something posted early in the month. For it to be one that posted the 19th? A bunch of people really like Frank and Ernest. That’s the only explanation.

Mercator-style map of the world, with the United States in the darkest pink and most of the Americas in a soft pink. Western Europe, Russia, India, China, and a fair bit of southeast Asia an Australia and New Zealand are also that uniform pink. Africa and the Middle East are grey, lacking readers.
You wouldn’t believe how long I spent trying to clean off the bit of monitor dust that was sitting there in the South Pacific before I worked out that it was Fiji. Well, you probably would; it was just a couple seconds and I worked it out by moving the window with the map on it.

There were 61 countries or country-like organizations to send me readers in May. There had been 54 countries for April and 59 for March. This past month 16 of them were single-reader countries. In April there were also 16 single-reader countries; in March, 17. Here’s the full roster:

Country Readers
United States 665
India 34
Canada 33
United Kingdom 31
Australia 19
Hong Kong SAR China 14
Germany 13
Mexico 10
France 8
South Korea 8
Nepal 7
New Zealand 7
Poland 7
Singapore 7
South Africa 7
Sweden 7
Chile 6
Denmark 6
Italy 6
Pakistan 5
Spain 5
Colombia 4
Panama 4
Slovenia 4
Algeria 3
Belize 3
Brazil 3
Egypt 3
Malaysia 3
Netherlands 3
Argentina 2
Bosnia & Herzegovina 2
China 2
Finland 2
Greece 2
Guam 2
Hungary 2
Ireland 2
Israel 2
Jamaica 2
Morocco 2
Norway 2
Peru 2
Thailand 2
Turkey 2
Austria 1
Bangladesh 1
Croatia 1
European Union 1 (*)
Fiji 1
Guatemala 1
Indonesia 1
Japan 1
Kuwait 1
Nigeria 1
Philippines 1
Puerto Rico 1
Russia 1
Taiwan 1
Uruguay 1
Vietnam 1

The European Union was the only single-reader country-like structure in May to have also been a single-reader place in April. None of the other countries have a streak going. Whoever my lone reader was in Jordan left after five months. The block of readers from Sweden has also dissipated but not disappeared altogether.

This year through the start of June I published 59 posts. This had a total of 57,871 words. This was 11,194 words published in May alone, for an average 1,119 words per post that month. My year-to-date average is 981 words per post. I’d been averaging 953 words per post at the start of May.

Through the start of June there’ve been 264 total likes of posts around here, an average of 4.5 likes per posting. That’s the same average likes per posting as the start of May saw. There’ve been a total of 91 comments, an average of 1.5 comments per posting. I notice, too, that this implies 17 comments in May, while the statistics panel I get claimed there were 12 comments in May. I think the discrepancy reflects pingbacks, one of my own posts referencing another. To verify this would need minutes of looking over the comments received here, though. So it’s sad to think of how this will never be done.

As of the start of June I’d posted 1,261 things here. They had a total of 78,957 page views from a 40,294 recorded unique visitors.

If you’d like to be a regular reader, there’s many ways to do it. One is to add my essays to your RSS reader, whatever that may be. If you do that, I will receive no statistics or logs or anything about your readership. It’ll be your secret. For a less-secret way, you can use the “Follow Nebusresearch” button, at the upper right corner of this page.

And if you follow me on Twitter as @Nebusj months will start with quality content like the above, of a couple pictures of a rabbit I saw from a parking lot. Thought you might like that.

Reading the Comics, May 30, 2019: Catching Out Tiger Mode


So this has been a week full of plans and machinations. But along the way, I made a discovery about Tiger. Curious? Of course you are. Who would not be? Read on and learn what my discovery is.

Hector D. Cantú and Carlos Castellanos’s Baldo for the 26th has Gracie counting by mathematical expressions. This kind of thing can be fun, at least for someone who enjoys doing arithmetic. Several years ago someone gave me a calendar in which every day was designated by an expression. As a mental exercise it wasn’t much, to my tastes. If you know that this is the second of the month, it’s no great work to figure out what \cos(0) + \sin(\frac{\pi}{2}) should be. But there is the fun in coming up with different ways to express a number. And here let me mention an old piece about how Paul Dirac worked out an expression for every counting number, using exactly four 2’s.

Gracie, little girl, jumping rope and counting: '4! 3 squared! 4 times 4! 20 percent of 210! Ounce in a half gallon!' Dad, to her aunt: 'Nobody counts their skips like Gracie.' Gracie: 'Degrees in a right angle!'
Hector D. Cantú and Carlos Castellanos’s Baldo for the 26th of May, 2019. It’s been a while since I’ve had reason to discuss this strip, but Baldo-inspired essays should be at this link.

John Graziano’s Ripley’s Believe It or Not for the 26th mentions several fairly believable things. The relevant part is about naming the kind of surface that a Pringles chip represents. That is, the surface a Pringles chip would be if it weren’t all choppy and irregular, and if it continued indefinitely.

The shape is, as Graziano’s Ripley’s claims, a hypberbolic paraboloid. It’s a shape you get to know real well if you’re a mathematics major. They turn up in multivariable calculus and, if you do mathematical physics, in dynamical systems. It’s also a shape mathematics majors get to calling a “saddle shape”, because it looks enough like a saddle if you’re not really into horses.

The shape is one of the “quadratic surfaces”. These are shapes which can be described as the sets of Cartesian coordinates that make a quadratic equation true. Equations in Cartesian coordinates will have independent variables x, y, and z, unless there’s a really good reason. A quadratic equation will be the sum of some constant times x, and some constant times x2, and some constant times y, and some constant times y2, and some constant times z, and some constant times z2. Also some constant times xy, and some constant times yz, and some constant times xz. No xyz, though. And it might have some constant added to the mix at the end of all this.

Trivias about a 155-year-old mousetrap which caught a mouse this year, the genus-species-subspecies designation for the Western Lowland Gorilla being 'gorilla gorilla gorilla', and that a Pringles shape is called a 'hyperbolic paraboloid'.
John Graziano’s Ripley’s Believe It or Not for the 26th of May, 2019. The collection of mathematics trivia I’ve noticed in Ripley’s Believe It Or Not should be at this link.

There are seventeen different kinds of quadratic surfaces. Some of them are familiar, like ellipsoids or cones. Some hardly seem like they could be called “quadratic”, like intersecting planes. Or parallel planes. Some look like mid-century modern office lobby decor, like elliptic cylinders. And some have nice, faintly science-fictional shapes, like hyperboloids or, as in here, hyperbolic paraboloids. I’m not a judge of which ones would be good snack shapes.

Horace reading a Math Quiz: 'Jack has 12 candy bars. He gives 10 to Jill. What does he have now?' Horace's answer; 'Jill's heart'.
Samson’s Dark Side of the Horse for the 26th of May, 2019. And I’m glad Horace has finally returned to these pages. Dark Side of the Horse gets discussed in essays at this link.

Samson’s Dark Side of the Horse for the 26th is a funny-answer-to-a-story-problem joke. I had thought these had all switched over to apples, rather than candy bars. But that would make the punch line less believable.

Bud Blake’s Tiger for the 31st is a rerun, of course. Blake died in 2005 and no one else drew his comic strip. It’s a funny-answer-to-a-story-problem joke. And, more, it’s a repeat of a Tiger strip I’ve already run here. I admit a weird pride when I notice a comic strip doing a repeat. It gives me some hope that I might still be able to remember things. But this is also a special Tiger repeat. It’s the strip which made me notice Bud Blake redrawing comics he had already used. This one is not a third iteration of the strip which reran in April 2015 and June 2016. It’s a straight repeat of the June 2016 strip.

Tiger, holding out his hands: 'If I had four apples in this hand ... and four more in this hand, what would I have?' Punkinhead: 'Really, really big giant hands!'
Bud Blake’s Tiger for the 31st of May, 2019. Appearances made by Tiger in these essays are at this link. Yes, I have to think about whether I mean to retire this link. But don’t worry: I’ll forget to act on that need.

The mystery to me now is why King Features apparently has less than three years’ worth of reruns in the bank for Tiger. The comic ran from 1965 to 2003, and it’s not as though the strip made pop culture references or jokes ripped from the headlines. Even if the strip changed its dimensions over the decades, to accommodate shrinking newspapers, there should be a decade at least of usable strips to rerun.

Man, handing a sheet to the Mathematician: 'Honey, your'e too pedantic. It's driving us apart. Here, I made a chart of how pedantic you've become.' She looks at the chart and sweats, more and more nervous. The last panel shows: it's an increasing trend, but the horizontal axis is labelled 'pedantry' and the vertical axis 'time'.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 31st of May, 2019. And as the Andertoons of multi-panel strips, Saturday Morning Breakfast Cereal features in the many essays at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 31st uses a chart to tease mathematicians, both in the comic and in the readership. The joke is in the format of the graph. The graph is supposed to argue that the Mathematician’s pedantry is increasing with time, and it does do that. But it is customary in this sort of graph for the independent variable to be the horizontal axis and the dependent variable the vertical. So, if the claim is that the pedantry level rises as time goes on, yes, this is a … well, I want to say wrong way to arrange the axes. This is because the chart, as drawn, breaks a convention. But convention is a tool to help people’s comprehension. We are right to ignore convention if doing so makes the chart better serve its purpose. Which, the punch line is, this does.


There’s just enough comics for me to do another essay this coming week. That next Reading the Comics post should be at this link around Thursday. That would be Tuesday except I need to fit my monthly readership report in sometime, don’t I? I think I need to, anyway.

Reading the Comics, May 25, 2019: Slighter Comics Edition.


It turned out to be Thursday. These things happen. The comics for the second half of last week were more marginal

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th is a joke about holographic cosmology, proving that there are such things as jokes about holographic cosmology. Cosmology is about the big picture stuff, like, why there is a universe and why it looks like that. It’s a rather mathematical field, owing to the difficulty of doing controlled experiments. Holograms are that same technology used back in the 80s to put shoddy three-dimensional-ish pictures of eagles on credit cards. (In the United States. I imagine they were other animals in other countries.) Holograms, at least when they’re well-made, encode the information needed to make a three-dimensional image in a two-dimensional surface. (Please pretend that anything made of matter is two-dimensional like that.)

Professor: '... therefore, we can explain our apparent three-dimensional universe as a hologram encoded in a two-dimensional field! You see, brothers and sisters? We were right all along!' Caption: 'Every so often, Professor Susskind sneaks into meetings of the Flat Earth Society to promote holographic cosmology.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th of May, 2019. Always glad to discuss Saturday Morning Breakfast Cereal, as you can see from these essays.

Holographic cosmology is a mathematical model for the universe. It represents the things in a space with a description of information on the boundary of this space. This seems bizarre and it won’t surprise you that key inspiration was in the strange physics of black holes. Properties of everything which falls into a black hole manifest in the event horizon, the boundary between normal space and whatever’s going on inside the black hole. The black hole is this three-dimensional volume, but in some way everything there is to say about it is the two-dimensional edge.

Dr Leonard Susskind did much to give this precise mathematical form. You didn’t think the character name was just a bit of whimsy, did you? Susskind’s work showed how the information of a particle falling into a black hole — information here meaning stuff like its position and momentum — turn into oscillations in the event horizon. The holographic principle argues this can be extended to ordinary space, the whole of the regular universe. Is this so? It’s hard to say. It’s a corner of string theory. It’s difficult to run experiments that prove very much. And we are stuck with an epistemological problem. If all the things in the universe and their interactions are equally well described as a three-dimensional volume or as a two-dimensional surface, which is “real”? It may seem intuitively obvious that we experience a three-dimensional space. But that intuition is a way we organize our understanding of our experiences. That’s not the same thing as truth.

Researcher one: 'Using simulated neural nets and quantum computing ... ' Researcher two: 'we've made a breakthrough in advanced AI. Behold.' One: 'Computer, two plus two equals five.' Computer: 'False. Two plus two equals four.' One, ready to yank the power cords out: 'Computer, two plus two equals five.' Computer: 'Correct, two plus two equals five.' Two: 'Adaptive reasoning, aka sense of self-preservation.' Duane: 'Impressive.'
Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde for the 22nd of May, 2019. Essays which mention some aspect of Barney and Clyde should appear at this link.

Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde for the 22nd is a joke about power, and how it can coerce someone out of truth. Arithmetic serves as an example of indisputable truth. It could be any deductive logic statement, or for that matter a definition. Arithmetic is great for the comic purpose needed here, though. Anyone can understand, at least the simpler statements, and work out their truth or falsity. And need very little word balloon space for it.

Caption: 'Why taco sauce? Why not steak sauce? Or Hollandaise? Barbecue?' Dingburg resident one: 'It's got to be taco sauce!' Dingburg resident two: 'Any other sauce would be sacrilegious!' Caption: 'But in an abandoned warehouse in Teaneck, New Jersey, a team of non-believers are at work!' One: 'This mix of duck sauce and salsa is just about ready!' Two: 'Piquant, yet chewy!' Caption: 'The new sauce gradually makes its way to Dingburg supermarkets, labelled Taco Sauce X-Treme.' Dingburger Three: 'After a swig, I feel all rationally ... ' Dingburger four: 'I think I just understood algebra!' Caption: 'An unexpected side effect of the new brew was a sudden ability to think logically for up to an hour after chugging a bottle.' Dingburger Five: 'Stop me before I rewrite the tax codes!'
Bill Griffith’s Zippy the Pinhead for the 25th of May, 2019. My attempts to form a quite rational and faintly linear discussion out of Zippy the Pinhead should be gathered here.

Bill Griffith’s Zippy the Pinhead for the 25th also features a quick mention of algebra as the height of rationality. Also as something difficult to understand. Most fields are hard to understand, when you truly try. But algebra works well for this writing purpose. Anyone who’d read Zippy the Pinhead has an idea of what understanding algebra would be like, the way they might not have an idea of holographic cosmology.

Two-bubble Venn diagram. The left bubble is 'Ryan Gosling', the right 'John Krasinski', and the intersection is 'Ryan Reynolds'. Caption: 'Menn Diagram'.
Teresa Logan’s Laughing Redhead Comics for the 25th of May, 2019. This one is a new tag. So there’s just the one Laughing Redhead Comics essay at this link. But that might change any day now!

Teresa Logan’s Laughing Redhead Comics for the 25th is the Venn diagram joke for the week, this one with a celebrity theme. Your choice whether the logic of the joke makes sense. Ryan Reynolds and John Krasinski are among those celebrities that I keep thinking I don’t know, but that it turns out I do know. Ryan Gosling I’m still not sure about.

And then there are a couple strips too slight even to appear in this collection. Dean Young and John Marshall’s Blondie on the 22nd did a lottery joke, with discussion of probability along the way. (And I hadn’t had a tag for ‘Blondie’ before, so that’s an addition which someday will baffle me.) Bob Shannon’s Tough Town for the 23rd mentions mathematics teaching. It’s in service of a pun.


And now I’ve had the past week covered. The next Reading the Comics post should be at this link come Sunday.

Reading the Comics, May 20, 2019: I Guess I Took A Week Off Edition


I’d meant to get back into discussing continuous functions this week, and then didn’t have the time. I hope nobody was too worried.

Bill Amend’s FoxTrot for the 19th is set up as geometry or trigonometry homework. There are a couple of angles that we use all the time, and they do correspond to some common unit fractions of a circle: a quarter, a sixth, an eighth, a twelfth. These map nicely to common cuts of circular pies, at least. Well, it’s a bit of a freak move to cut a pie into twelve pieces, but it’s not totally out there. If someone cuts a pie into 24 pieces, flee.

Offscreen voice: 'So a pizza sliced into fourths has ... ' Paige: '90 degrees per slice.' Voice: 'Correct! And a pizza sliced into sixths has ... ' Page: '60 degrees per slice.' Voice: 'Good! And a pizza sliced into eighths has ... ' Paige: '45 degrees per slice.' Voice: 'Yep! I'd say you're ready for your geometry final, Paige.' Paige: 'Woo-hoo!' Voice, revealed to be Peter: 'Now help me clean up these [ seven pizza ] boxes.' Page: 'I still don't understand why teaching me this required *actual* pizzas.'
Bill Amend’s FoxTrot for the 19th of May, 2019. Essays featuring FoxTrot, either the current (Sunday-only) strips or the 1990s-vintage reruns, should be at this link.

Tom Batiuk’s vintage Funky Winkerbean for the 19th of May is a real vintage piece, showing off the days when pocket electronic calculators were new. The sales clerk describes the calculator as having “a floating decimal”. And here I must admit: I’m poorly read on early-70s consumer electronics. So I can’t say that this wasn’t a thing. But I suspect that Batiuk either misunderstood “floating-point decimal”, which would be a selling point, or shortened the phrase in order to make the dialogue less needlessly long. Which is fine, and his right as an author. The technical detail does its work, for the setup, by existing. It does not have to be an actual sales brochure. Reducing “floating point decimal” to “floating decimal” is a useful artistic shorthand. It’s the dialogue equivalent to the implausibly few, but easy to understand, buttons on the calculator in the title panel.

Calculator salesman: 'This little pocket calculator is a real beauty. It's nice and light so you can take it anywhere. It has an eight-digit readout with automatic roundoff. Not only that, but it has a floating decimal which enables you to solve ANY type of problem with it!' Les Moore: 'Amazing! May I try it out?' (To the calculator) 'Hello, pocket calculator? Why do I have so much trouble getting girls to like me?'
Tom Batiuk’s vintage Funky Winkerbean for the 19th of May, 2019. The strip originally ran the 17th of June, 1973. Comics Kingdom is printing both the current Funky Winkerbean strips and early-70s reprints. Essays that mention Funky Winkerbean, old or new, should appear at this link.

Floating point is one of the ways to represent numbers electronically. The storage scheme is much like scientific notation. That is, rather than think of 2,038, think of 2.038 times 103. In the computer’s memory are stored the 2.038 and the 3, with the “times ten to the” part implicit in the storage scheme. The advantage of this is the range of numbers one can use now. There are different ways to implement this scheme; a common one will let one represent numbers as tiny as 10-308 or as large as 10308, which is enough for most people’s needs.

The disadvantage is that floating point numbers aren’t perfect. They have only around (commonly) sixteen digits of significance. That is, the first sixteen or so nonzero numbers in the number you represent mean anything; everything after that is garbage. Most of the time, that trailing garbage doesn’t hurt. But most is not always. Trying to add, for example, a tiny number, like 10-20, to a huge number, like 1020 won’t get the right answer. And there are numbers that can’t be represented correctly anyway, including such exotic and novel numbers as \frac{1}{3} . A lot of numerical mathematics is about finding ways to compute that avoid these problems.

Back when I was a grad student I did have one casual friend who proclaimed that no real mathematician ever worked with floating point numbers, because of the limitations they impose. I could not get him to accept that no, in fact, mathematicians are fine with these limitations. Every scheme for representing numbers on a computer has limitations, and floating point numbers work quite well. At some point, you have to suspect some people would rather fight for a mistaken idea they already have than accept something new.

Matrix-O-Magic: Draw a nine-square grid on a notepad, filling in the numbers 1-9 like this: 2, 9, 4 // 7, 5, 3 // 6, 1, 8 Hand the pad and marker to a friend and tell him to pick any row of three numbers, upward, downward, or diagonal. Tell him to black out any numbers not in his row. Instruct your friend to add up his three randomly chosen numbers. Ask your friend to flip through the rest of the notepad to make sure the pages are blank. All the pages are blank except one. That one bears the number that his numbers added up to: 15. (All the rows/columns/diagonals add to 15; because the other numbers are blacked out your friend won't notice. If asked to do the trick more than once the grid can be made to look different by rotating the order of the numbers left or right, et, 6, 7, 2 // 1, 5, 9 // 8, 3, 4.)
Mac King and Bill King’s Magic in a Minute for the 19th of May, 2019. So far as I know all these panels are new ones, although they do reuse gimmicks now and then. But the arithmetic and logic tricks featured in Magic In A Minute get discussed at this link, when they get mention from me at all.

Mac King and Bill King’s Magic in a Minute for the 19th does a bit of stage magic supported by arithmetic: forecasting the sum of three numbers. The trick is that all eight possible choices someone would make have the same sum. There’s a nice bit of group theory hidden in the “Howdydoit?” panel, about how to do the trick a second time. Rotating the square of numbers makes what looks, casually, like a different square. It’s hard for human to memorize a string of digits that don’t have any obvious meaning, and the longer the string the worse people are at it. If you’ve had a person — as directed — black out the rows or columns they didn’t pick, then it’s harder to notice the reused pattern.

The different directions that you could write the digits down in represent symmetries of the square. That is, geometric operations that would replace a square with something that looks like the original. This includes rotations, by 90 or 180 or 270 degrees clockwise. Mac King and Bill King don’t mention it, but reflections would also work: if the top row were 4, 9, 2, for example, and the middle 3, 5, 7, and the bottom 8, 1, 6. Combining rotations and reflections also works.

If you do the trick a second time, your mark might notice it’s odd that the sum came up 15 again. Do it a third time, even with a different rotation or reflection, and they’ll know something’s up. There are things you could do to disguise that further. Just double each number in the square, for example: a square of 4/18/8, 14/10/6, 12/2/16 will have each row or column or diagonal add up to 30. But this loses the beauty of doing this with the digits 1 through 9, and your mark might grow suspicious anyway. The same happens if, say, you add one to each number in the square, and forecast a sum of 18. Even mathematical magic tricks are best not repeated too often, not unless you have good stage patter.

Wavehead, to classmate, over lunch: 'Did you know that every square is a rhombus, but not every rhombus is a square? I mean, you can't make this stuff up!'
Mark Anderson’s Andertoons for the 20th of May, 2019. Always glad to discuss Andertoons, as you can see from these essays.

Mark Anderson’s Andertoons for the 20th is the Mark Anderson’s Andertoons for the week. Wavehead’s marveling at what seems at first like an asymmetry, about squares all being rhombuses yet rhombuses not all being squares. There are similar results with squares and rectangles. Still, it makes me notice something. Nobody would write a strip where the kid marvelled that all squares were polygons but not all polygons were squares. It seems that the rhombus connotes something different. This might just be familiarity. Polygons are … well, if not a common term, at least something anyone might feel familiar. Rhombus is a more technical term. It maybe never quite gets familiar, not in the ways polygons do. And the defining feature of a rhombus — all four sides the same length — seems like the same thing that makes a square a square.


There should be another Reading the Comics post this coming week, and it should appear at this link. I’d like to publish it Tuesday but, really, Wednesday is more probable.

Reading the Comics, May 16, 2019: Two and Two Edition


It might be more fair to call this a blackboard edition, as three of the strips worth discussing feature that element. But I think I’ve used that name recently. And two of the strips feature specifically 2 + 2, so I’ll use that instead.

And here’s a possible movie heads-up. Turner Classic Movies, United States feed, is showing Monday at 9:30 am (Eastern/Pacific) All-American Chump. All I know about this 1936 movie is from its Leonard Maltin review:

[ Stuart ] Erwin is funny, in his usual country bumpkin way, as a small-town math whiz known as “the human adding machine” who is exploited by card sharks and hustlers. Fairly diverting double-feature item.

People with great powers of calculation were — and still are — with us. Before calculating machines were common they were, pop mathematicians tell us, in demand for doing the kinds of arithmetic mathematicians and engineers need a lot of. They’d also have value in performing, if they can put together some good patter. And, sure, gambling is just another field that needs calculation done well. I have no idea the quality of the film (it’s rated two and a half stars, but Leonard Maltin rates many things two and a half stars). But it’s there if you’re curious. The film also stars Robert Armstrong. I assume it’s not the guy I know but, you know? We live in a strange world. Now on to the comics.

Glenn McCoy and Gary McCoy’s The Flying McCoys for the 13th uses the image of a blackboard full of mathematics symbols to represent deep thought. The equations on the board are mostly nonsense, although some, like E = mc^2 , have obvious meaning. Many of the other symbols have some meaning to them too. In the upper-right corner, for example, is what looks like E = \hbar \omega . This any physics major would recognize: it’s the energy of a photon, which is equal to Planck’s constant (that \hbar stuff) times its frequency.

Pair of scientist or mathematician types standing in front of a blackboard full of symbols. One says to the other: 'You're overthinking this.'
Glenn McCoy and Gary McCoy’s The Flying McCoys for the 13th of May 2019. I had thought I was just writing about this strip, and no, yesterday I posted a mention that I was not writing about it. You can see that and other mentions of The Flying McCoys at this link.

And there are other physics-relevant symbols. In the bottom center is a line that starts \oint \vec{B} . The capital B is commonly used to represent a magnetic field. The arrow above the capital B is a warning that this is a vector, which magnetic fields certainly are. (Mathematicians see vectors as a quite abstract concept. Physicists are more likely to see them as an intensity and direction, like forces, and the fields that make fields.) The \oint symbol comes from vector calculus. It represent an integral taken along a closed loop, a shape that goes out along some path and comes back to where it started without crossing itself. This turns out to be useful all the time in dynamics problems. So the McCoys drew something that doesn’t mean anything, but looks ready to mean things.

“Overthinking this” is a problem common to mathematicians, even at an advanced level. Real problems don’t make clear what their boundaries are, the things that are important and the things that aren’t and the things that are convenient but not essential. Making mistakes picking them out, and working too hard on the wrong matters, will happen.

Chesney, cat: 'Question: Are there more stars in the sky or grains of sand on all the beach of the world?' Annie: 'I would say definitely stars.' Jack, cat: 'That settles it.' Chesney, walking away, to Jack: 'Sand it is.' Jack: 'She's never right.'
Graham Harrop’s Ten Cats for the 14th of May 2019. This is a new tag; a strip about a girl and the ten cats she caretakes doesn’t get much into mathematics. I did mention it once, years ago, before I was tagging essays by comic strip name. Anyway, this and future mentions of Ten Cats should be at this link.

Graham Harrop’s Ten Cats for the 14th sees the cats pondering the counts of vast things. These are famous problems. Archimedes composed a text, The Sand Reckoner, which tried to estimate how much sand there could be in the universe. To work on the question he had to think of new ways to represent numbers. Grains of sand become numerous by being so tiny. Stars become numerous by the universe being so vast. Comparing the two quantities is a good challenge. For both numbers we have to make estimates. The volume of beaches in the world. The typical size of a grain of sand. The number of galaxies in the universe. The typical number of stars in a galaxy. There’s room to dispute all these numbers; we really have to come up with a range of possible values, with maybe some idea of what seems more likely.

Student, waving to two adults behind him, and explaining to his nonplussed arithmetic teacher (the blackboard has a big 2 + 2 = on it): 'This is a mathematics professor and my attorney. They'll explain why my answer is technically and legally correct.'
Thaves’s Frank and Ernest for the 15th of May 2019. Essays which mention some aspect of Frank and Ernest should appear at this link.

Thaves’s Frank and Ernest for the 15th has the student bringing authority to his answer. The mathematician is called on to prove an answer is “technically” correct. I’m not sure whether the kid is meant to be prefacing the answer he’s about to give, or whether his answer was rewriting the horizontal “2 + 2 = ” in a vertical form.

Serf: 'How much would you charge to draw up my will?' Lawyer: 'I take one-third of your estate.' Serf: 'That's impossible.' Lawyer: 'Why?' (The serf's children enter.) Serf: 'Three don't go into *five*.'
Brant Parker and Johnny Hart’s The Wizard of Id Classics for the 15th of May 2019. By the way the lawyer’s name is Larsen E Pettifogger, in case that ever comes up. I don’t know that the Serf has a name. This particular strip is from 1969. Essays inspired by something in either new-run Wizard of Id strips or 50-year-old repeats should be at this link.

Brant Parker and Johnny Hart’s The Wizard of Id Classics for the 15th is built around the divisibility of whole numbers, and of relative primes. Setting the fee as some simple integer fraction of the whole has practicality to it. It likely seemed even more practical in the days before currencies decimalized. The common £sd style currency Europeans used before decimals could be subdivided many ways evenly, with one-third of a pound (livre, Reichsgulden, etc) becoming 80 pence (deniers, Pfennig, etc). Unit fractions, and combinations of unit fractions, could offer interesting ways to slice up anything to a desired amount.

Kid, to the teacher, after answering the blackboard's 2+2 as 5: 'I make mistakes to keep you on your toes!'
Jim Unger’s Herman for the 16th of May 2019. This strip is also a repeat; Unger retired from regularly drawing new strips in 1992, and died in 2012. (From 1997 he occasionally updated old ones or drew new ones.) I have no idea when this first ran. The strip gets little attention from me, but the essays — with this, two of them — mentioning Herman should appear at this link.

Jim Unger’s Herman for the 16th is a student-talking-back-to-the-teacher strip. It also uses the 2 + 2 problem. It’s a common thing for teachers to say they learn from their students. It’s even true, although I son’t know that people ever quite articulate how teachers learn. A good mistake is a great chance to learn. A good mistake shows off a kind of brilliant twist. That the student has understood some but not all of the idea, and has filled in the misunderstood parts with something plausible enough one has to think about why it’s wrong. And why someone would think the wrong idea might be right. There is a kind of mistake that inspires you to think closely about what “right” has to be, and students who know how to make those mistakes are treasures.


And for comic strips that aren’t quite worth a paragraph. Julia Kaye’s Up and Out for the 13th uses mathematics as stand-in for the sort of general education that anybody should master. David Waisglass and Gordon Coulthart’s Farcus for the 17th I don’t think is trying to be a mathematics joke. It’s sufficient joke that the painter’s spelled ‘sign’ wrong. But it did hit on the spelling that would encourage mathematics teachers to notice the strip. Patrick Roberts’s Todd the Dinosaur for the 18th mentions sudoku.


And with that I am caught up on the past week’s mathematically-themed comic strips. My next Reading the Comics post should be next Sunday, and at this link. Oh, I could have made the edition name something bragging about being on time.

Reading the Comics, May 11, 2019: I Concede I Am Late Edition


I concede I am late in wrapping up last week’s mathematically-themed comics. But please understand there were important reasons for my not having posted this earlier, like, I didn’t get it written in time. I hope you understand and agree with me about this.

Bill Griffith’s Zippy the Pinhead for the 9th brings up mathematics in a discussion about perfection. The debate of perfection versus “messiness” begs some important questions. What I’m marginally competent to discuss is the idea of mathematics as this perfect thing. Mathematics seems to have many traits that are easy to think of as perfect. That everything in it should follow from clearly stated axioms, precise definitions, and deductive logic, for example. This makes mathematics seem orderly and universal and fair in a way that the real world never is. If we allow that this is a kind of perfection then … does mathematics reach it?

Sluggo: 'You don't mess with perfection, Zippy.' Zippy: 'Sluggo, are you saying you're perfect?' Sluggo: 'Yes. I am perfect.' Zippy: 'And I am perfect too, Sluggo, huh?' Sluggo: 'No. You are not perfect. Your lines and your circles are irregular and messy.' Zippy: 'Which is better, Sluggo, perfect or messy?' Sluggo: 'Perfect is better. In a fight between messy and perfect, Sluggo always kills Zippy!' Zippy: 'In lieu of a human sacrifice, please accept this perfect parallelogram!'
Bill Griffith’s Zippy the Pinhead for the 9th of May, 2019. I am surprised to learn this is not a new tag. Essays discussing Zippy the Pinhead are at this link. Ernie, here, is Ernie Bushmiller, creator and longtime artist and writer for Nancy. He’s held in regard by some of the art community for his economic and streamlined drawing and writing style. You might or might not like his jokes, but you can’t deny that he made it easy to understand what was supposed to be funny and why it was supposed to be. It’s worth study if you like to know how comic strips can work.

Even the idea of a “precise definition” is perilous. If it weren’t there wouldn’t be so many pop mathematics articles about why 1 isn’t a prime number. It’s difficult to prove that any particular set of axioms that give us interesting results are also logically consistent. If they’re not consistent, then we can prove absolutely anything, including that the axioms are false. That seems imperfect. And few mathematicians even prepare fully complete, step-by-step proofs of anything. It takes ridiculously long to get anything done if you try. The proofs we present tend to show, instead, the reasoning in enough detail that we’re confident we could fill in the omitted parts if we really needed them for some reason. And that’s fine, nearly all the time, but it does leave the potential for mistakes present.

Zippy offers up a perfect parallelogram. Making it geometry is of good symbolic importance. Everyone knows geometric figures, and definitions of some basic ideas like a line or a circle or, maybe, a parallelogram. Nobody’s ever seen one, though. There’s never been a straight line, much less two parallel lines, and even less the pair of parallel lines we’d need for a parallellogram. There can be renderings good enough to fool the eye. But none of the lines are completely straight, not if we examine closely enough. None of the pairs of lines are truly parallel, not if we extend them far enough. The figure isn’t even two-dimensional, not if it’s rendered in three-dimensional things like atoms or waves of light or such. We know things about parallelograms, which don’t exist. They tell us some things about their shadows in the real world, at least.

Business Man on phone: 'A trillion is still a stop-and-think decision for me.'
Mark Litzler’s Joe Vanilla for the 9th of May, 2019. And this one is just barely not a new tag. This and other essays mentioning Joe Vanilla should be at this link.

Mark Litzler’s Joe Vanilla for the 9th is a play on the old joke about “a billion dollars here, a billion dollars there, soon you’re talking about real money”. As we hear more about larger numbers they seem familiar and accessible to us, to the point that they stop seeming so big. A trillion is still a massive number, at least for most purposes. If you aren’t doing combinatorics, anyway; just yesterday I was doing a little toy problem and realized it implied 470,184,984,576 configurations. Which still falls short of a trillion, but had I made one arbitrary choice differently I could’ve blasted well past a trillion.

A Million Monkeys At A Million Typewriters. Scene of many monkeys on typewriters. One pauses, and thinks, and eventually pulls a bottle of liquor from the desk, thinking: 'But how can I credibly delay Hamlet's revenge until Act V?'
Ruben Bolling’s Super-Fun-Pak Comix for the 9th of May, 2019. This one I knew wouldn’t be a new tag, not with how nerdy-but-in-the-good-ways Ruben Bolling writes. Essays mentioning Super-Fun-Pak Comix are at this link.

Ruben Bolling’s Super-Fun-Pak Comix for the 9th is another monkeys-at-typewriters joke, that great thought experiment about probability and infinity. I should add it to my essay about the Infinite Monkey Theorem. Part of the joke is that the monkey is thinking about the content of the writing. This doesn’t destroy the prospect that a monkey given enough time would write any of the works of William Shakespeare. It makes the simple estimates of how unlikely that is, and how long it would take to do, invalid. But the event might yet happen. Suppose this monkey decided there was no credible way to delay Hamlet’s revenge to Act V, and tried to write accordingly. Mightn’t the monkey make a mistake? It’s easy to type a letter you don’t mean to. Or a word you don’t mean to. Why not a sentence you don’t mean to? Why not a whole act you don’t mean to? Impossible? No, just improbable. And the monkeys have enough time to let the improbable happen.

A big wobbly scribble. Caption; 'Eric the Circle in the 20th dimension, where shape has no meaning.'
Eric the Circle, this by Kingsnake, for the 10th of May, 2019. I keep figuring to retire Eric the Circle as it seems to be all in reruns. But then I keep finding strips that, as far as I can tell, I haven’t discussed before. Essays about stuff raised by Eric the Circle should be at this link.

Eric the Circle for the 10th, this one by Kingsnake, declares itself set in “the 20th dimension, where shape has no meaning”. This plays on a pop-cultural idea of dimensions as a kind of fairyland, subject to strange and alternate rules. A mathematician wouldn’t think of dimensions that way. 20-dimensional spaces — and even higher-dimensional spaces — follow rules just as two- and three-dimensional spaces do. They’re harder to draw, certainly, and mathematicians are not selected for — or trained in — drawing, at least not in United States schools. So attempts at rendering a high-dimensional space tend to be sort of weird blobby lumps, maybe with a label “N-dimensional”.

And a projection of a high-dimensional shape into lower dimensions will be weird. I used to have around here a web site with a rotatable tesseract, which would draw a flat-screen rendition of what its projection in three-dimensional space would be. But I can’t find it now and probably it ran as a Java applet that you just can’t get to work anymore. Anyway, non-interactive videos of this sort of thing are common enough; here’s one that goes through some of the dimensions of a tesseract, one at a time. It’ll give some idea how something that “should” just be a set of cubes will not look so much like that.

Hayden: 'I don't need to know long division because there's a calculator on my phone.' Dustin: 'What happens if someday you don't have a phone?' Hayden: 'Then I've got problems long division won't solve.'
Steve Kelly and Jeff Parker’s Dustin for the 11th of May, 2019. And, you know, this strip is just out there, doing its business. Essays about some topic raised by Dustin are at this link.

Steve Kelly and Jeff Parker’s Dustin for the 11th is a variation on the “why do I have to learn this” protest. This one is about long division and the question of why one needs to know it when there’s cheap, easily-available tools that do the job better. It’s a fair question and Hayden’s answer is a hard one to refute. I think arithmetic’s worth knowing how to do, but I’ll also admit, if I need to divide something by 23 I’m probably letting the computer do it.


And a couple of the comics that week seemed too slight to warrant discussion. You might like them anyway. Brian Boychuk and Ron Boychuk’s Chuckle Brothers for the 5th featured a poorly-written numeral. Charles Schulz’s Peanuts Begins rerun for the 6th has Violet struggling with counting. Glenn McCoy and Gary McCoy’s The Flying McCoys for the 8th has someone handing in mathematics homework. Henry Scarpelli and Craig Boldman’s Archie rerun for the 9th talks about Jughead sleeping through mathematics class. All routine enough stuff.


This and other Reading the Comics posts should appear at this link. I mean to have a post tomorrow, although it might make my work schedule a little easier to postpone that until Monday. We’ll see.

Are These Numbers A Thing?


A friend recently had a birthday. His fun way of mentioning his age was to put it as a story problem, as in this tweet.

He refined the question eventually. His age was now a palindrome, but it had just been a prime number. I came back with the obvious answer: he’s 268,862 years old.

And he was curious how I came up with that. Specifically how I ended up in the six-digit range. The next “age” after 44 would be 212. Jumping to a number that’s outside the range of plausible human ages is the obvious joke. How I got to six digits is less obvious. But it seemed to me that if three digits is funny, then four digits must be funnier. The four-digit palindromes-succeeding-a-prime seemed boring. Five digits it is, then. Oh, but then there’s the thousands comma and that’s not in the middle of the number. Six digits it is. This gives you insight into why I am a humor blogger and not a successful humor blogger.

I didn’t go looking numbers myself, by the way. I had Octave, this open-source clone of Matlab, tell me whether numbers were prime. I just had to think of palindromic numbers.

There are a couple of plausible human ages that are palindromes-succeeding-a-prime. At least if you accept a one-digit number as a palindrome. (I don’t see a reason not to.) Those would be 3, 4, 6, 8, and 44. After that we get into numbers that humans are not likely to reach, such as 212 and 242 and 252 and 272 and 282. Then nothing until 434, 444, 464, and so on. Certainly nothing in the 500’s.

So that’s got me wondering two things, and they’re open questions. The first is, is this sequence a thing? That is, has anybody done any kind of study about palindromes-after-a-prime? I’m not saying that this is an important sequence. This is a sequence that you look at and say, “Huh” about. But there are a lot of sequences that you can mostly just say “Huh” about. That doesn’t mean we don’t know anything about them. I checked in the Online Encyclopedia of Integer Sequences, and found nothing. But I’m not confident in my searching ability.

The second thing is what anyone studying this sequence would first like to know. Is this an infinitely long sequence? Or is there a largest palindrome-succeeding-a-prime? My instinct is to say there’s not a largest. There are infinitely many prime numbers. There are infinitely many palindromic numbers. Surely the coincidence that a prime is followed by a palindrome happens infinitely often. That is purely a guess, however.

There could be and end to this. Consider truncatable primes, a prime number which (in base ten) is still prime if you truncate any string of its leftmost or its rightmost digits. That is, like, chop either any number of digits off the left end, or off the right end, of 3797, and you still have a prime number. There are only finitely many primes that let you do this. Specifically there are 15 (base-ten) primes that let you chop off either the left or the right side and still have a prime.

Bizarrely, if you allow only chopping off digits on the left side, there are 4,260 left-truncatable primes. If you allow only chopping off digits on the right side, there are 83. If you chop off alternating digits, one from the left side and then one from the right, then there are 920,720,315 such primes. This hasn’t anything to do with my friend’s numbers. They’re just an example that this sort of sequence can Peter out, and in unpredictable ways. Wouldn’t you have guessed there to be about as many left-truncatable as right-truncatable numbers? And fewer alternating-truncatable numbers than either? … Well, I would have, anyway.

So I don’t know. And I know that number theory problems like this have a habit of being either solvable by a tiny bit of cleverness or being basically impossible to do. No idea which this is, but someone out there might enjoy passing a dull meeting by doodling integers.

Reading the Comics, May 8, 2019: Strips With Art I Like Edition


Of course I like all the comics. … Well, that’s not literally true; but I have at least some affection for nearly all of the syndicated comics. This essay I bring up some strips, partly, because I just like them. This is my content hole. If you want a blog not filled with comic strips, go start your own and don’t put these things on it.

Mark Anderson’s Andertoons for the 5th is the Mark Anderson’s Andertoons for the week. Also a bit of a comment on the ability of collective action to change things. Wavehead is … well, he’s just wrong about making the number four plus the number four equal to the number seven. Not based on the numbers we mean by the words “four” and “seven”, and based on the operation we mean by “plus” and the relationship we mean by “equals”. The meaning of those things is set by, ultimately, axioms and deductive reasoning and the laws of deductive reasoning and there’s no changing the results.

Wavehead, to another student: 'If I say 4 + 4 = 7 it's wrong. If you say 4 + 4 = 7 it's wrong. But if the entire first grade says 4 + 4 = 7, well, now she has to take us seriously.
Mark Anderson’s Andertoons for the 5th of May, 2019. Essays mentioning Andertoons are at this link and also at nearly every Reading the Comics post, it feels like.

But. The thing we’re referring to when we say “seven”? Or when we write the symbol “7”? That is convention. That is a thing we’ve agreed on as a reference for this concept. And that we can change, if we decide we want to. We’ve done this. Look at a thousand-year-old manuscript and the symbol that looks like ‘4’ may represent the number we call five. And the names of numbers are just common words. They’re subject to change the way every other common word is. Which is, admittedly, not very subject. It would be almost as much bother to change the word ‘four’ as it would be to change the word ‘mom’. But that’s not impossible. Just difficult.

Viivi: 'Oh, sorry! Oh, pain!' Wagner: 'Stop worrying. 85% of fears never come through.' Viivi: 'That means 15% do! It's worse than I thought.'
Juba’s Viivi and Wagner for the 5th of May, 2019. I don’t often have chances to talk about Viivi and Wagner but when I do, it’s here.

Juba’s Viivi and Wagner for the 5th is a bit of a percentage joke. The characters also come to conclude that a thing either happens or it does not; there’s no indefinite states. This principle, the “excluded middle”, is often relied upon for deductive logic, and fairly so. It gets less clear that this can be depended on for predictions of the future, or fears for the future. And real-world things come in degrees that a mathematical concept might not. Like, your fear of the home catching fire comes true if the building burns down. But it’s also come true if a quickly-extinguished frying pan fire leaves the wall scorched, embarrassing but harmless. Anyway, relaxing someone else’s anxiety takes more than a quick declaration of statistics. Show sympathy.

Dogs in school. The dog teacher is pointing to '1 + 1' on the blackboard. A dog student whispers to the other, 'Sometimes I feel so stupid.'
Harry Bliss and Steve Martin’s Bliss for the 6th of May, 2019. Yes, by the way, it’s the Steve Martin you know and love from Looney Tunes: Back In Action and from the 1996 Sergeant Bilko movie. Anyway I haven’t had chance to write about this strip before but this and future appearances of Bliss should be here.

Harry Bliss and Steve Martin’s Bliss for the 6th is a cute little classroom strip, with arithmetic appearing as the sort of topic that students feel overwhelmed and baffled by. It could be anything, but mathematics uses the illustration space efficiently. The strip may properly be too marginal to include, but I like Bliss’s art style and want more people to see it.

Spud: 'It's official, Wallace. My socks are *too* tight. And I know it'll take at least three minutes to run home and change. Yet I can see the bus is only two stops away.' Wallace: 'I can stall for a good thirty seconds.' Spud: 'My life is a sadistic math problem.'
Will Henry’s Wallace the Brave for the 7th of May, 2019. This is one of the comic strips I’m most excited about, the last several years. Wallace the Brave appears in essays at this link.

Will Henry’s Wallace the Brave for the 7th puts up what Spud calls a sadistic math problem. And, well, it is a story problem happening in their real life. You could probably turn this into an actual exam problem without great difficulty.

Ruthie, holding up a triangle: 'What's this shape?' James: 'A square!' Ruthie: 'I already *told* you what it is, James! You're just acting dumb to hurt my feelings! Stop it! N-n-now (sob) what does this look like to you? (Sniff)?' James: 'A cryangle!'
Rick Detorie’s One Big Happy for the 8th of May, 2019. There are two strings of One Big Happy available for daily reading. Appearances by the current or the several-years-old GoComics prints of One Big Happy should be at this link.

Rick Detorie’s One Big Happy for the 8th is a bit of wordplay built around geometry, as Ruthie plays teacher. She’s a bit dramatic, but she always has been.


I’ll read some more comics for later in this week. That essay, and all similar comic strip talk, should appear at this link. Thank you.

Why I’ll Say 1/x Is A Continuous Function And Why I’ll Say It Isn’t


So let me finally follow up last month’s question. That was whether the function “\frac{1}{x} ” is continuous. My earlier post lays out what a mathematician means by a “continuous function”. The short version is, we have a good definition for a function being continuous at a point in the domain. If it’s continuous at every point in the domain, it’s a continuous function.

The definition of continuous-at-a-point has some technical stuff that I’m going to skip this essay. The important part is that the stuff ordinary people would call “continuous” mathematicians agree with. Like, if you draw a curve representing the function without having to lift your pen off the paper? That function’s continuous. At least the stretch you drew was.

So is the function “\frac{1}{x} ” continuous? What if I said absolutely it is, because ‘x’ is a number that happens to be … oh, let’s say it’s 3. And \frac{1}{3} is a constant function; of course that’s continuous. Your sensible response is to ask if I want a punch in the nose. No, I do not.

One of the great breakthroughs of algebra was that we could use letters to represent any number we want, whether or not we know what number it is. So why can’t I get away with this? And the answer is that we live in a society, please. There are rules. At least, there’s conventions. They’re good things. They save us time setting up problems. They help us see things the current problem has with other problems. They help us communicate to people who haven’t been with us through all our past work. As always, these rules are made for our convenience, and we can waive them for good reason. But then you have to say what those reasons are.

What someone expects, if you write ‘x’ without explanation it’s a variable and usually an independent one. Its value might be any of a set of things, and often, we don’t explicitly know what it is. Letters at the start of the alphabet usually stand for coefficients, some fixed number with a value we don’t want to bother specifying. In making this division — ‘a’, ‘b’, ‘c’ for coefficients, ‘x’, ‘y’, ‘z’ for variables — we are following Réné Descartes, who explained his choice of convention quite well. And there are other letters with connotations. We tend to use ‘t’ as a variable if it seems like we’re looking at something which depends on time. If something seems to depend on a radius, ‘r’ goes into service. We use letters like ‘f’ and ‘g’ and ‘h’ for functions. For indexes, ‘i’ and ‘j’ and ‘k’ get called up. For total counts of things, or for powers, ‘n’ and ‘m’, often capitalized, appear. The result is that any mathematician, looking at the expression

\sum_{j = i}^{n} a_i f(x_j)

would have a fair idea what kinds of things she was looking at.

So when someone writes “the function \frac{1}{x} ” they mean “the function which matches ‘x’, in the domain, with \frac{1}{x} , in the range”. We write this as “f(x) = \frac{1}{x} ”. Or, if we become mathematics majors, and we’re in the right courses, we write “f:x \rightarrow \frac{1}{x} ”. It’s a format that seems like it’s overcomplicating things. But it’s good at emphasizing the idea that a function can be a map, matching a set in the domain to a set in the range.

This is a tiny point. Why discuss it at any length?

It’s because the question “is \frac{1}{x} a continuous function” isn’t well-formed. There’s important parts not specified. We can make it well-formed by specifying these parts. This is adding assumptions about what we mean. What assumptions we make affect what the answer is.

A function needs three components. One component is a set that’s the domain. One component is a set that’s the range. And one component is a rule that pairs up things in the domain with things in the range. But there are some domains and some ranges that we use all the time. We use them so often we end up not mentioning them. We have a common shorthand for functions which is to just list the rule.

So what are the domain and range?

Barring special circumstances, we usually take the domain that offers the most charitable reading of the rule. What’s the biggest set on which the rule makes sense? The domain is that. The range we find once we have the domain and rule. It’s the set that the rule maps the domain onto.

So, for example, if we have the function “f(x) = x2”? That makes sense if ‘x’ is any real number. if there’s no reason to think otherwise, we suppose the domain is the set of all real numbers. We’d write that as the set R. Whatever ‘x’ is, though, ‘x2‘ is either zero or a positive number. So the range is the real numbers greater than or equal to zero. Or the nonnegative real numbers, if you prefer.

And even that reasonably clear guideline hides conventions. Like, who says this should be the real numbers? Can’t you take the square of a complex-valued number? And yes, you absolutely can. Some people even encourage it. So why not use the set C instead?

Convention, again. If we don’t expect to need complex-valued numbers, we don’t tend to use them. I suspect it’s a desire not to invite trouble. The use of ‘x’ as the independent variable is another bit of convention. An ‘x’ can be anything, yes. But if it’s a number, it’s more likely a real-valued number. Same with ‘y’. If we want a complex-valued independent variable we usually label that ‘z’. If we need a second, ‘w’ comes in. Writing “x2” alone suggests real-valued numbers.

And this might head off another question. How do we know that ‘x’ is the only variable? How do we know we don’t need an ordered pair, ‘(x, y)’? This would be from the set called R2, pairs of real-valued numbers. It uses only the first coordinate of the pair, but that’s allowed. How do we know that’s not going on? And we don’t know that from the “x2” part. The “f(x) = ” part gives us that hint. If we thought the problem needed two independent variables, it would usually list them somewhere. Writing “f(x, y) = x2” begs for the domain R2, even if we don’t know what good the ‘y’ does yet. In mapping notation, if we wrote “f:(x, y) \rightarrow x^2 ” we’d be calling for R2. If ‘x’ and ‘z’ both appear, that’s usually a hint that the problem needs coordinates ‘x’, ‘y’, and ‘z’, so that we’d want R3 at least.

So that’s the maybe frustrating heuristic here. The inferred domain is the smallest biggest set that the rule makes sense on. The real numbers, but not ordered pairs of real numbers, and not complex-valued numbers. Something like that.

What does this mean for the function “f(x) = \frac{1}{x} ”? Well, the variable is ‘x’, so we should think real numbers rather than complex-valued ones. There no ‘y’ or ‘z’ or anything, so we don’t need ordered sets. The domain is something in the real numbers, then. And the formula “\frac{1}{x} ” means something for any real number ‘x’ … well, with the one exception. We try not to divide by zero. It raises questions we’d rather not have brought up.

So from this we infer a domain of “all the real numbers except 0”. And this in turn implies a range of “all the real numbers except 0”.

Is “f(x) = \frac{1}{x} ” continuous on every point in the domain? That is, whenever ‘x’ is any real number besides zero? And, well, it is. A proper proof would be even more heaps of paragraphs, so I’ll skip it. Informally, you know if you drew a curve representing this function there’s only one point where you would ever lift your pen. And that point is 0 … which is not in this domain. So the function is continuous at every point in the domain. So the function’s continuous. Done.

And, I admit, not quite comfortably done. I feel like there’s some slight-of-hand anyway. You draw “\frac{1}{x} ” and you absolutely do lift your pen, after all.

So, I fibbed a little above. When I said the range was “the set that the rule maps the domain onto”. I mean, that’s what it properly is. But finding that is often too much work. You have to find where the function would be its smallest, which is often hard, or at least tedious. You have to find where it’s largest, which is just as tedious. You have to find if there’s anything between the smallest and largest values that it skips. You have to find all these gaps. That’s boring. And what’s the harm done if we declare the range is bigger than that set? If, for example, we say the range of’ x2‘ is all the real numbers, even though we know it’s really only the non-negative numbers?

None at all. Not unless we’re taking an exam about finding the smallest range that lets a function make sense. So in practice we’ll throw in all the negative numbers into that range, even if nothing matches them. I admit this makes me feel wasteful, but that’s my weird issue. It’s not like we use the numbers up. We’ll just overshoot on the range and that’s fine.

You see the trap this has set up. If it doesn’t cost us anything to throw in unneeded stuff in the range, and it makes the problem easier to write about, can we do that with the domain?

Well. Uhm. No. Not if we’re doing this right. The range can have unneeded stuff in it. The domain can’t. It seems unfair, but if we don’t set hold to that rule, we make trouble for ourselves. By ourselves I mean mathematicians who study the theory of functions. That’s kind of like ourselves, right? So there’s no declaring that “\frac{1}{x} ” is a function on “all” the real numbers and trusting nobody to ask what happens when ‘x’ is zero.

But we don’t need for a function’s rule to a be a single thing. Or a simple thing. It can have different rules for different parts of the domain. It’s fine to declare, for example, that f(x) is equal to “\frac{1}{x} ” for every real number where that makes sense, and that it’s equal to 0 everywhere else. Or that it’s 1 everywhere else. That it’s negative a billion and a third everywhere else. Whatever number you like. As long as it’s something in the range.

So I’ll declare that my idea of this function is an ‘f(x)’ that’s equal to “\frac{1}{x} ” if ‘x’ is not zero, and that’s equal to 2 if ‘x’ is zero. I admit if I weren’t writing for an audience I’d make ‘f(x)’ equal to 0 there. That feels nicely symmetric. But everybody picks 0 when they’re filling in this function. I didn’t get where I am by making the same choices as everybody else, I tell myself, while being far less successful than everybody else.

And now my ‘f(x)’ is definitely not continuous. The domain’s all the real numbers, yes. But at the point where ‘x’ is 0? There’s no drawing that without raising your pen from the paper. I trust you’re convinced. Your analysis professor will claim she’s not convinced, if you write that on your exam. But if you and she were just talking about functions, she’d agree. Since there’s one point in the domain where the function’s not continuous, the function is not continuous.

So there we have it. “\frac{1}{x} ”, taken in one reasonable way, is a continuous function. “\frac{1}{x} ”, taken in another reasonable way, is not a continuous function. What you think reasonable is what sets your answer.

Reading the Comics, May 4, 2019: Wednesday Looks A Lot Like Tuesday Edition


I didn’t get this published on Tuesday, owing to circumstances beyond my control, such as my not writing it Monday. I have hopes of catching up on all the writing I want to do. Someday, I might.

Marcus Hamilton and Scott Ketcham’s Dennis the Menace for the 2nd hardly seems like Dennis lives up to his “Menace” title. It seems more like he’s discovered wordplay. This is usually no worse than “mildly annoying”. Joey seems alarmed, but I must tell you, reader, he’s easily alarmed. But I think there is some depth here.

Dennis, sitting beside some papers and crayons and his friend: 'When it comes to numbers, Joey ... there's always *one more*.'
Marcus Hamilton and Scott Ketcham’s Dennis the Menace for the 2nd of May, 2019. This is twice in three months that this venerable comic’s made an appearance here. Who saw that coming? This and past appearances of Dennis the Menace are at this link. Future appearances should be, too, if they happen.

One is that, as we’ve thought of counting numbers, there is always “one more”. This doesn’t have to be. We could work with perfectly good number systems that have a largest number. We do, in fact. Every computer programming language has some largest integer that it will deal with. If you need a larger number, you have to do something clever. Your clever idea will let you address some range of bigger numbers, but it too will have a maximum. We’ve set those limits large enough that, usually, they’re not an inconvenience. They’re still there.

But those limits are forced on us by the many failings of matter. What when we get just past Plato’s line’s division, into the reasoning of pure mathematics? There we can set up counting numbers. The standard way to do this is to suppose there is a number “1”. And to suppose that, for any counting number we have, there is a successor, a number one-plus-that. If Joey were to ask why there has to be, all Dennis could do is shrug. This makes an axiom out of there always being one more. If you don’t like it, make some other arithmetic. Anyway we only understand any of this using fallible matter, so good luck.

This progression can be heady, though. The counting numbers are probably the most understandable infinitely large set there is. Thinking about them seriously can induce the sort of dizzy awe that pondering Deep Time or the vastness of space can do. That seems a bit above Dennis’s age level, but some people are stricken with the infinite sooner than others are.

Charlie Brown: 'You think I'm dumb, don't you? Well, ask me a question! Ask me anything!' Patty: 'All right, how much is two and two?' Charlie Brown: 'Hmmm ... Why don't you ask me something more practical?'
Charles Schulz’s Peanuts Begins rerun for the 2nd of May, 2019. The strip originally ran the 23rd of March, 1951. When I have reason to discuss Peanuts, either the current-newspaper-reruns or these early-50s reruns, the essays should appear here.

Charles Schulz’s Peanuts Begins rerun for the 2nd has Charlie Brown dismiss arithmetic as impractical. It fits the motif of mathematics as an unworldly subject. There’s the common joke that pure mathematics even dreams of being of no use to anyone. Arithmetic, though, has always been a practical subject. It introduces us to many abstract ideas, particularly group theory. This subject looks at what we can do with systems that work like arithmetic without necessarily having numbers, or anything that works with numbers.

Venn diagram labelled 'Caffeine Routine'. One circle, beside a cup of coffee, is labelled 'me'. The second circle, beside a travel mug of coffee, is labelled 'also me'. The intersection is labelled 'too much coffee'.
John Atkinson’s Wrong Hands for the 3rd of May, 2019. Other essays featuring by Wrong Hands are at this link.

John Atkinson’s Wrong Hands for the 3rd is the Venn Diagram joke for the week. I’m not sure the logic of the joke quite holds up, but it’s funny at a glance and that’s as much as it needs to do.

Several geometric figures lay on the beach. A triangle, wearing sunglasses, says to an un-worn hat and pair of glasses beside it: 'Ahh, this is the life, eh, Vera? ... Vera?' Caption: 'Bermuda Triangles'.
Scott Hilburn’s The Argyle Sweater for the 4th of May, 2019. The many essays discussing The Argyle Sweater appear at this link.

Scott Hilburn’s The Argyle Sweater for the 4th is the anthropomorphic geometric figures joke for the week.


And a couple of comic strips mentioned mathematics, although in too slight a way to discuss. Dana Simpson’s Phoebe and her Unicorn on the 30th of April started a sequence in which doodles on Phoebe’s homework came to life. That it’s mathematics homework was mostly incidental. I’m open to the argument that mathematics encourages doodling in a way that, say, spelling does not. I’d also be open to the argument you aren’t doing geometry if you don’t doodle. Anyway. Dan Thompson’s Brevity for the 2nd of May features Sesame Street’s Count von Count. It’s a bit of wordplay on the use of “numbers” for songs. And, of course, the folkloric tradition of vampires as compulsive counters.


With that, I’m temporarily caught up on my comics. I’m falling behind almost every week, though. Come Sunday, the next essay should appear here.

Reading the Comics, May 1, 2019: Not Perfectly Certain Edition


There’s several comics from the first half of last week that I can’t perfectly characterize. They seem to be on-topic enough for my mathematical discussions. It’s just how exactly they are on-topic that I haven’t quite got. Some weeks are like that.

Dave Whamond’s Reality Check for the 28th circles around being a numerals joke. It’s built on the binary representation of numbers that we’ve built modern computers on. And on the convention that “(Subject) 101” is the name for an introductory course in a subject. This convention of course numbering — particularly, three-digit course numbers, with the leading digit representing the year students are expected to take it — seems to have spread in American colleges in the 1930s. It’s a compromise, as many things are. As college programs of study become more specialized there’s the need for a greater number of courses in each field. And there’s a need to give people some hint of the course level. “Numerical Methods” could be a sophomore, senior, or grad-student course; how should someone from a different school know what to expect? But the pull of the serial number, and the idea that ’01’ must be the first in a field, is hard to resist.

Woman looking at college classroom doors: English 101 and then Computers 101 0101001 101010100 0110110011 100100100.
Dave Whamond’s Reality Check for the 28th of April, 2019. Essays which include Reality Check should be gathered at this link.

Anyway, the long string of zeroes and ones after the original ‘101’ is silliness and that’s all it has to be. The number one-hundred-and-one in binary would be a mere “1100101”, which doesn’t start with the important one-oh-one, and isn’t a big enough string of digits to be funny. Maybe this is a graduate course. The number given, if we read it as a single long binary number, would be 182,983,026,468. I’ve been to schools which use four-digit course codes. Twelve digits seems excessive.

Man carrying a large numeral 8 walks to a man sitting on a recliner in a field. The man with the eight asks, 'Is this nine?' The sitting man says, 'No. Eight.' The man with the eight says, 'I've got the wrong number'. And he walks away, carrying the 8, in a silent final panel.
John Deering’s Strange Brew for the 29th of April, 2019. Other appearances by Strange Brew are at this link.

John Deering’s Strange Brew for the 29th circles around being an anthropomorphic numerals joke. At least it is a person using a large representation of the number eight. I’m not sure how to characterize it, or why I find the strip amusing. It’s a strange one.

Cop, presenting a handcuffed 2, 3, 5, and 7 to his supervisor at the Vice Squad: 'We're breaking up the numbers racket. These are some of the prime suspects.'
Thaves’s Frank and Ernest for the 1st of May, 2019. This is the first time in maybe a month that I’ve written about Thaves’s strip. But last time, and other earlier appearances, of Frank and Ernest should be gathered at this link.

Thaves’s Frank and Ernest for the 1st is, finally, a certain anthropomorphic numerals joke. With wordplay about prime numbers being unavoidably prime suspects. … And when I was a kid, I had no idea what “numbers rackets” were, other than a thing sometimes mentioned on older sitcoms. That it involved somehow literally taking numbers and doing … something … that the authorities didn’t like was mysterious. I don’t remember what surely hilarious idea the young me had for what that might even mean. I suspect that, had I seen this strip at the time, I would have understood this wasn’t really whatever was going on. But I would have explained to my parents what a prime number was, and they would put up with my doing so, because that’s just what our relationship was.

Fish standing at a podium with a laptop, and behind, a screen with a string of overlapping circles. It says to the other: 'These are Venn diagras for my presentation. And no, I'm not tooting.'
Dave Whamond’s Reality Check for the 1st of May, 2019. And it’s a bit odd to have two Reality Check strips in the same essay. But I’m glad to have the strip back at all, since I discovered I had somehow lost the comic for a couple of months.

Dave Whamond’s Reality Check for the 1st is more or less the Venn Diagram joke for this essay. It’s a bit of a fourth-wall-breaking strip: the joke wouldn’t really work from the other goldfish’s perspective. Anyway, only two of those figures are proper Venn diagrams. The topmost figure, with five circles, and the bottommost, with three, aren’t proper Venn diagrams. Only some of the possible intersections between sets exist there. They are proper Euler diagrams, though.

Person giving a presentation: 'As this slide indicates, the most popular pies are apple, chocolate, and pumpkin.' The slide is a pie chart, showing (in decreasing popularity) apple, chocolate, pumpkin, pecan, cherry, lemon meringue, coconut cream, banana cream, key lime, and blueberry. Caption; 'Metadata'.
Wayno’s WaynoVision for the 1st of May, 2019. This appears to be the first time I’ve mentioned this comic. Well. This and any future essays mentioning WaynoVision should be at this link.

Wayno’s WaynoVision for the 1st is the pie-chart joke for the essay. It’s not as punchy as that Randolph Itch strip I kept bringing back around. But it’s on the same theme, mixing the metaphor of the pie chart with literal pies.


There’s one more Reading the Comics post before I’ve got all last week’s strips covered. That, I hope to have published and available at this link for Tuesday.

How April 2019 Treated My Mathematics Blog


Well, I deserved that. After a fair start April pretty well flopped for me: the last two weeks of the month I ended up not writing any of the things I should have. If it weren’t for reblogs and heads-up posts I wouldn’t have even reached ten posts for the month. I’m not sure when I’ve posted that little. It looks like sometime early 2014.

So April was my least-red month in a long while. Since December 2017, looks like. But of the things within my control, post count and schedule are the things that most affect readership. And boy was April a writer-blocked month for me. Here’s how bad it was.

The strikingly uniform monthly readership of my blog for the past four and a half years; there's a drop in April. 1,020 views, 667 visitors, 1.53 views per visitor, and 12 posts published.
Oh, so, how I got results going back this far. WordPress has this page that offers site owners a review of their readership figures. The URL for it looks like https://wordpress.com/stats/month/nebusresearch.wordpress.com with the name of whatever your particular WordPress blog is on the end there. If you add to the URL ?startDate=2016-12-01 — or some other starting date, in the format YYYY-MM-DD — then you get just what you’d imagine. What I haven’t figured is how to set the range. You’d think adding an endDate property would do it, and no, it does’t. But there’s little left- and right-arrows to the either side of the ‘Stats for (Given Month and Year)’ and clicking the right-arrow there will expand the range. It really doesn’t want to show more than about four and a half years in one screen, though.

So I still broke a thousand page views; I haven’t fallen below that since the depressing month of December 2017. I admit part of why I pushed that what-grade-you-need post on Monday was that I was a little short of a thousand views and hoped to get above that. March 2019 had 1,391 views, and February 1,275. In April there were 668 unique visitors, my lowest since July 2018 (also with 668) or February 2018 (611) depending on how you count “lowest”. There’d been 954 unique visitors in March and 835 in February.

The number of likes went back to its plummet in April: only 40 things liked at all around here. In March there’d been 97 likes; in February 44. And here’s where fiddling with the startDate property really hurts, because there has been this incredible secular decline in likes. I mean, in all 2015 I never dropped below 179 likes in one month, and never below 107 in 2016. In 2017 the minimum was 70. In 2018 the minimum was 37. I don’t know what’s making me less likable.

Comments were up in April, although they’d almost have to be. There were 14 in April; March saw only four. February had ten. I might do another A To Z just to get people talking to me.

Well, here’s the roster of popular essays this past month:

That’s not a bad spread of posts.

Mercator-style projection map of the world. The United States is shown in the deepest red. The rest of North and South America, mostly, is in pink; as are western Europe, the Baltics, South Asia, Russia, and Australia.
Your Victoria II challenge of the month: create this Empire with client states. Easy mode: you may start as the United Kingdom, any Indian nation, Russia, or Brazil instead. Hard mode: accomplish this by 1900.

54 countries sent me readers at all this past month. 16 of them were single-reader countries. That’s down from the 59 countries of March and 17 single-reader countries. Also from February’s 73 countries and 20 single-reader countries. But here’s the country roster:

Country Readers
United States 688
Canada 44
United Kingdom 39
Sweden 26
India 22
Australia 19
Pakistan 12
Brazil 11
France 10
Italy 9
Malaysia 9
Singapore 8
Norway 7
Slovenia 7
Belgium 6
Germany 6
Hong Kong SAR China 6
Russia 6
Spain 6
Austria 5
Philippines 5
Saudi Arabia 5
South Africa 5
United Arab Emirates 5
Finland 4
Greece 4
South Korea 4
Denmark 3
Japan 3
Nepal 3
Vietnam 3
Chile 2
Hungary 2
Israel 2
Jamaica 2
Switzerland 2
Thailand 2
Turkey 2
Bolivia 1
Costa Rica 1
Djibouti 1
European Union 1
Ghana 1
Guam 1
Ireland 1
Jordan 1 (****)
Macedonia 1
Mexico 1
Netherlands 1
Peru 1 (**)
Serbia 1
U.S. Virgin Islands 1
Ukraine 1
Venezuela 1

Peru’s been a single-reader country for three months now. Jordan’s been one for five months. That’s the only ongoing streak. I don’t know what’s got so many Swedish readers in lately. I fear there might have been a misunderstanding somewhere.

This year, through the start of May I’ve posted 49 pieces. This has gotten a total of 46,677 words, according to whatever definition of ‘word’ WordPress uses. This is 9,943 words in April, which for me counts as laconic. The average post length this year has dropped to 953 words, down from the 993 at the start of April. There were twelve posts in April, technically, for an average of 829 words per post. There’ve been 221 total likes for the year, putting me at an average of 4.5 likes per post. At the start of April there had been an average of 4.9 likes per post. This year there’ve been a total of 74 comments, for an average of 1.5 comments per posting.

Or so says WordPress. But my post about March 2019 said I’d reached 52 comments by the end of March, and I’d had 14 comments in April. Something isn’t adding up here. I get these yearly totals from the Insights panel, and I wonder if that’s counting pingbacks — one WordPress post linking to another — as comments. Those aren’t counted in the monthly-comments-total mentioned above.

May starts with my having made 1,251 posts in total. These have attracted overall 77,976 page views from an acknowledged 39,572 unique visitors.

I’m always glad to have you as visitor. If you’d like to add my little efforts here to your WordPress Reader, please use the “Follow Nebusresearch” button at the upper right corner of this page. I confess despairing a bit at recent followers. Not you, of course, kind reader. But a lot of recent followers seem to be those curious blogs that just reblog articles about search-engine-optimization and that’s … all … a baffling exercise to me. Anyway, if you want to follow me without showing up in my or anyone’s analytics, good on you. Here is the RSS feed for my essays. And to get back to the surveillance, I’m @Nebusj on Twitter. And to encourage you to follow: I’m trying to start each month with a pair of rabbit pictures. Not to brag, but this month’s?

You’re welcome.

Reading the Comics, April 26, 2019: Absurd Equation Edition


And now I’ll cover the handful of comic strips which ran last week and which didn’t fit in my Sunday report. And link to a couple of comics that ultimately weren’t worth discussion in their own right, mostly because they were repeats of ones I’ve already discussed. I have been trimming rerun comics out of my daily reading. But there are ones I like too much to give up, at least not right now.

Bud Blake’s Tiger for the 25th has Tiger quizzing Punkinhead on counting. The younger kid hasn’t reached the point where he can work out numbers without a specific physical representation. It would come, if he were in one of those comics where people age.

Tiger: 'What comes after eleven?' Punkinhead: 'I can't do it. I don't have enough fingers to count on!' Tiger, handing a baseball glove: 'Use this.'
Bud Blake’s Tiger for the 25th of April, 2019. Essays that bring up something in Tiger appear at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 24th is an optimization problem, and an expectation value problem. The wisdom-seeker searches for the most satisfying life. The mathematician-guru offers an answer based in probability and expectation values. List all the possible outcomes, and how probable each are, and how much of the relevant quantity you get (or lose) with each outcome. This is a quite utilitarian view of life-planning. Finding the best possible outcome, given certain constraints, is another big field of mathematics.

Woman seeking enlightenment: 'Should human being strive for pleasure or fulfillment?' Mathematician guru: 'That's a math question, not a philosophy question. Life of pleasure: probability of success 80%, life satisfaction is 5 on scale of 0 to 10; weighted value is 0.8 * 5 = 4. Life of fulfilment: probability of success is 20%, satisfaction is 10; weighted value is 0.2 * 10 = 2.' 'So no life strategy gets you even halfway to the maximum value?' 'There is one. Muddle through: probability of success is 100%. Life satisfaction if successful is 7. 7 * 1.0 = 7.' Woman: 'I tell you, we are here on Earth to ---- around. Kurt Vonnegut.' Mathematician: 'Did you know he trained as a scientist before writing books?'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 24th of April, 2019. There’s plenty of discussion of Saturday Morning Breakfast Cereal at this link.

John Atkinson’s Wrong Hands for the 26th is a nonsense-equation panel. It’s built on a cute idea. If you do wan to know how many bears you can fit in the kitchen you would need something like this. Not this, though. You can tell by the dimensions. ‘x’, as the area of the kitchen, has units of, well, area. Square feet, or square meters, or square centimeters, or whatever is convenient to measure its area. The average volume of a bear, meanwhile, has units of … volume. Cubic feet, or cubic meters, or cubic centimeters, or the like. The one divided by the other has units of one-over-distance.

Powerpoint-style slide: Impractical Equation 1. Number of bears you can fit in your kitchen. (x / y) x d = ... x: area of your kitchen. y: average volume of a bear. d: desire to have bears in your kitchen.
John Atkinson’s Wrong Hands for the 26th of April, 2019. Other essays featuring by Wrong Hands are at this link.

And I don’t know what the units of desire to have bears in your kitchen are, but I’m guessing it’s not “bear-feet”, although that would be worth a giggle. The equation would parse more closely if y were the number of bears that can fit in a square foot, or something similar. I say all this just to spoil Atkinson’s fine enough bit of nonsense.

Skippy: 'I can run ten miles in 3520 seconds flat!' Sooky: 'How do ya know?' Skippy: ''Cause I ran fifty yards an' timed myself.'
Percy Crosby’s Skippy rerun for the 26th of April, 2019. It first ran the 3rd of December, 1931. This and other mentions of Crosby’s brilliant Skippy should appear at this link.

Percy Crosby’s Skippy for the 26th is a joke built on inappropriate extrapolation. 3520 seconds is a touch under an hour. Skippy’s pace, if he could keep it up, would be running a mile every five minutes, 52 seconds. That pace isn’t impossible — I find it listed on charts for marathon runners. But that would be for people who’ve trained to be marathon or other long-distance runners. They probably have different fifty-yard run times.


And now for some of the recent comics that didn’t seem worth their own discussion, and why they didn’t.

Niklas Eriksson’s Carpe Diem for the 20th features reciting the digits of π as a pointless macho stunt. There are people who make a deal of memorizing digits of π. Everyone needs hobbies, and memorizing meaningless stuff is a traditional fanboy’s way of burying oneself in the thing appreciated. Me, I can give you π to … I want to say sixteen digits. I might have gone farther in my youth, but I was heartbroken when I learned one of the digits I had memorized I got wrong, and so after correcting that mess I gave up going farther.

Rick Detorie’s One Big Happy rerun for the 22nd has Ruthie seeking mathematics help from the homework hotline. The mathematics is just a pretext. And Richard Thompson’s Richard’s Poor Almanac for the 22nd is the color version of that comic with the Platonic Fir tree, discussed several times. Bud Fisher’s Mutt and Jeff for the 25th reprints the pre-relettering version of >the eating-the-roast-beef joke This is the strip that I’d found changed to “eating ham” in 2018, part of the strip’s mysterious and unexplained relettering.


And now I am, briefly, caught up on the comic strips. I’ll be behind again by Sunday, though. I’ll do something about that, in an essay you should be able to find at this link.

What Does It Take To Get A C This Class?


I’m posting this for several sordid reasons. First is that I want to test whether WordPress has changed something in how pingbacks — a post linking to another post — get handled. Second is I want to get my post count for the month up from its pitifully low number. I’m at something like negative four posts for all April. Third is that oh, yes, it is about that time of the semester when a kind of student is trying to study just hard enough to get a 79.6 percent in their classwork. So they want to study up to an 86.2 on the final and not waste their efforts studying up to an 86.5.

So here’s a couple tables I set up years ago. They show, for some common breakdowns of how much the final exam is worth, and what your class average is before going into the finals, what you’d need to get a 60, 65, 70, 80, or 90.

If your case isn’t handled in the above examples, here’s an essay with the complete formula needed to handle any circumstance, including extra credit.

But seriously you can’t study yourself up to “just” enough to get your target grade for the course. Study to understand the subject and take the grade as it is.

Reading the Comics, April 24, 2019: Mic Drop Edition Edition


I can’t tell you how hard it is not to just end this review of last week’s mathematically-themed comic strips after the first panel here. It really feels like the rest is anticlimax. But here goes.

John Deering’s Strange Brew for the 20th is one of those strips that’s not on your mathematics professor’s office door only because she hasn’t seen it yet. The intended joke is obvious, mixing the tropes of the Old West with modern research-laboratory talk. “Theoretical reckoning” is a nice bit of word juxtaposition. “Institoot” is a bit classist in its rendering, but I suppose it’s meant as eye-dialect.

Cowboys at the 'Institoot of Theoretical Reckoning'. One at the whiteboard says, 'Well, boys, looks like this here's the end of the line!' The line is a long string of what looks like legitimate LaTeX
John Deering’s Strange Brew for the 20th of April, 2019. Other appearances by Strange Brew, including ones less diligent about making the blackboard stuff sensible, are at this link.

What gets it a place on office doors is the whiteboard, though. They’re writing out mathematics which looks legitimate enough to me. It doesn’t look like mathematics though. What’s being written is something any mathematician would recognize. It’s typesetting instructions. Mathematics requires all sorts of strange symbols and exotic formatting. In the old days, we’d handle this by giving the typesetters hazard pay. Or, if you were a poor grad student and couldn’t afford that, deal with workarounds. Maybe just leave space in your paper and draw symbols in later. If your university library has old enough papers you can see them. Maybe do your best to approximate mathematical symbols using ASCII art. So you get expressions that look something like this:

  / 2 pi  
 |   2
 |  x cos(theta) dx - 2 F(theta) == R(theta)
 |
/ 0

This gets old real fast. Mercifully, Donald Knuth, decades ago, worked out a great solution. It uses formatting instructions that can all be rendered in standard, ASCII-available text. And then by dark incantations and summoning of Linotype demons, re-renders that as formatted text. It handles all your basic book formatting needs — much the way HTML, used for web pages, will — and does mathematics much more easily. For example, I would enter a line like:

\int_{0}^{2\pi} x^2 \cos(\theta) dx - 2 F(\theta) \equiv R(\theta)

And this would be rendered in print as:

\int_{0}^{2\pi} x^2 \cos(\theta) dx - 2 F(\theta) \equiv R(\theta)

There are many, many expansions available to this, to handle specialized needs, hardcore mathematics among them.

Anyway, the point that makes me realize John Deering was aiming at everybody with an advanced degree in mathematics ever with this joke, using a string of typesetting instead of the usual equations here?

The typesetting language is named TeX.

Wavehead, at lunch: 'You know if I were the other shapes I'd be like, 'listen, circle, you can have a perimeter or a circumference, but you can't have both'.'
Mark Anderson’s Andertoons for the 21st of April, 2019. When do I ever not discuss this comic? All the essays at this link are about Andertoons, at least in part.

Mark Anderson’s Andertoons for the 21st is the Mark Anderson’s Andertoons for the week. It’s about one of those questions that nags at you as a kid, and again periodically as an adult. The perimeter is the boundary around a shape. The circumference is the boundary around a circle. Why do we have two words for this? And why do we sound all right talking about either the circumference or the perimeter of a circle, while we sound weird talking about the circumference of a rhombus? We sound weird talking about the perimeter of a rhombus too, but that’s the rhombus’s fault.

The easy part is why there’s two words. Perimeter is a word of Greek origin; circumference, of Latin. Perimeter entered the English language in the early 15th century; circumference in the 14th. Why we have both I don’t know; my suspicion is either two groups of people translating different geometry textbooks, or some eager young Scholastic with a nickname like ‘Doctor Magnifico Triangulorum’ thought Latin sounded better. Perimeter stuck with circules early; every etymology I see about why we use the symbol π describes it as shorthand for the perimeter of the circle. Why `circumference’ ended up the word for circles or, maybe, ellipses and ovals and such is probably the arbitrariness of language. I suspect that opening “circ” sound cues people to think of it for circles and circle-like shapes, in a way that perimeter doesn’t. But that is my speculation and should not be mistaken for information.

Information panel about numerals, including a man who typed every number from one to a million, using one finger; it took over 16 years, seven months. Puzzle: add together every number that, written as a word, consists of three letters; what's the total?
Steve McGarry’s KidTown for the 21st of April, 2019. It’s rare that this panel is on-topic enough for me to bring up, but at least a few KidTown panels are discussed here.

Steve McGarry’s KidTown for the 21st is a kids’s information panel with a bit of talk about representing numbers. And, in discussing things like how long it takes to count to a million or a billion, or how long it would take to type them out, it gets into how big these numbers can be. Les Stewart typed out the English names of numbers, in words, by the way. He’d also broken the Australian record for treading water, and for continuous swimming.

Bub: 'I don't like crosswords because I'm not good at word stuff. I'm much better at math. That's why I like sudoku.' Betty: 'What math? There's no adding or subtracting or multiplying in sudoku.' Bub: 'That's my favorite kind of math.' Betty: 'If you were better at word stuff, you'd know you're confusing math with logic.'
Gary Delainey and Gerry Rasmussen’s Betty for the 24th of April, 2019. I don’t seem to have discussed this comic before. This and future appearances by Betty should be at this link.

Gary Delainey and Gerry Rasmussen’s Betty for the 24th is a sudoku comic. Betty makes the common, and understandable, conflation of arithmetic with mathematics. But she’s right in identifying sudoku as a logical rather than an arithmetic problem. You can — and sometimes will see — sudoku type puzzles rendered with icons like stars and circles rather than numerals. That you can make that substitution should clear up whether there’s arithmetic involved. Commenters at GoComics meanwhile show a conflation of mathematics with logic. Certainly every mathematician uses logic, and some of them study logic. But is logic mathematics? I’m not sure it is, and our friends in the philosophy department are certain it isn’t. But then, if something that a recognizable set of mathematicians study as part of their mathematics work isn’t mathematics, then we have a bit of a logic problem, it seems.


Come Sunday I should have a fresh Reading the Comics essay available at this link.