Reading the Comics, November 11, 2018: November 11, 2018 Edition


There were just enough mathematically-themed comic strips last week to make two editions for this coming week. All going well I’ll run the other half on either Wednesday or Thursday. There is a point that isn’t quite well, which is that one of the comics is in dubious taste. I’ll put that at the end, behind a more specific content warning. In the meanwhile, you can find this and hundreds of other Reading the Comics posts at this link.

Thaves’s Frank and Ernest for the 11th is wordplay, built on the conflation of “negative” as in numbers and “negative” as in bad. I’m not sure the two meanings are unrelated. The word ‘negative’ itself derives from the Latin word meaning to deny, which sounds bad. It’s easy to see why the term would attach to what we call negative numbers. A number plus its negation leaves us zero, a nothing. But it does make the negative numbers sound like bad things to have around, or to have to deal with. The convention that a negative number is less than zero implies that the default choice for a number is one greater than zero. And the default choice is usually seen as the good one, with everything else a falling-away. Still, -7 is as legitimate a number as 7 is; it’s we who think one is better than another.

Alien Frank: 'The first Earthling election confused me. I expected campaign signs with things like '-5 + -2'.' Alien Ernest: 'The term is 'negative ads', not 'negative adds'.' Frank: 'I thought the pier would be crowded with people casting ballots. I heard there are voting machines so I expected to see a line of robots waiting at the polls. At least there were no natural disasters. I was worried about actual landslides because of all the mudslinging.'
Thaves’s Frank and Ernest for the 11th of November, 2018. Other essays mentioning Frank and Ernest will be at this link.

J C Duffy’s Lug Nuts for the 11th has the Dadaist panel present prime numbers as a way to communicate. I suspect Duffy’s drawing from speculations about how to contact alien intelligences. One problem with communicating with the truly alien is how to recognize there is a message being sent. A message too regular will look like a natural process, one conveying no more intelligence than the brightness which comes to most places at dawn and darkness coming at sunset. A message too information-packed, peculiarly, looks like random noise. We need an intermediate level. A signal that it’s easy to receive, and that is too hard to produce by natural processes.

Caption: 'Don's first, primitive attempt at communication was limited to prime numbers.' Don, speaking to an angered woman: '2 ... 3 ... 5 ... 7 ... 11 ... 13 ... 17 ...'
J C Duffy’s Lug Nuts for the 11th of November, 2018. This and other essays mentioning Lug Nuts will be at this link.

Prime numbers seem like a good compromise. An intelligence that understands arithmetic will surely notice prime numbers, or at least work out quickly what’s novel about this set of numbers once given them. And it’s hard to imagine an intelligence capable of sending or receiving interplanetary signals that doesn’t understand arithmetic. (Admitting that yes, we might be ruling out conversational partners by doing this.) We can imagine a natural process that sends out (say) three pulses and then rests, or five pulses and rests. Or even draws out longer cycles: two pulses and a rest, three pulses and a rest five pulses and a rest, and then a big rest before restarting the cycle. But the longer the string of prime numbers, the harder it is to figure a natural process that happens to hit them and not other numbers.

We think, anyway. Until we contact aliens we won’t really know what it’s likely alien contact would be like. Prime numbers seem good to us, but — even if we stick to numbers — there’s no reason triangular numbers, square numbers, or perfect numbers might not be as good. (Well, maybe not perfect numbers; there aren’t many of them, and they grow very large very fast.) But we have to look for something particular, and this seems like a plausible particularity.

Lucy: 'Charlie Brown, how much is zero times zero?' Charlie Brown: 'Zero.' Lucy: 'ZERO? Oh come on, Charlie Brown, it's *got* to be *something*. I'll put down three. That sounds just about right. 'Zero', he says ... ha!' Charlie Brown: 'Things like that make my stomach hurt.'
Charles Schulz’s Peanuts Begins for the 11th of November, 2018. It originally ran the 11th of August, 1954. Essays discussing topics raised by Peanuts will be at this link. That’s for either the “current” newspaper run, currently doing strips from 1971, or for the “vintage” reruns as here, showing strips from 1954.

Charles Schulz’s Peanuts Begins for the 11th is an early strip, from the days when Lucy would look to Charlie Brown for information. And it’s a joke built on conflating ‘zero’ with ‘nothing’. Lucy’s right that zero times zero has to be something. That’s how multiplication works. That the number zero is something? That’s a tricky concept. I think being mathematically adept can blind one to how weird that is. If you’re used to how zero is the amount of a thing you have to have nothing of that thing, then we start to see what’s weird about it.

But I’m not sure the strip quite sets that up well. I think if Charlie Brown had answered that zero times zero was “nothing” it would have been right (or right enough) and Lucy’s exasperation would have flowed more naturally. As it is? She must know that zero is “nothing”; but then why would she figure “nothing times nothing” has to be something? Maybe not; it would have left Charlie Brown less reason to feel exasperated or for the reader to feel on Charlie Brown’s side. Young Lucy’s leap to “three” needs to be at least a bit illogical to make any sense.

Now to the last strip and the one I wanted to warn about. It alludes to gun violence and school shootings. If you don’t want to deal with that, you’re right. There’s other comic strips to read out there. And this for a comic that ran on the centennial of Armistice Day, which has to just be an oversight in scheduling the (non-plot-dependent) comic.

Continue reading “Reading the Comics, November 11, 2018: November 11, 2018 Edition”

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Reading the Comics, November 9, 2018: Standing For Things Edition


There was something in common in two of the last five comic strips worth attention from last week. That’s good enough to give the essay its name.

Greg Cravens’s The Buckets for the 8th showcases Toby discovering the point of letters in algebra. It’s easy to laugh at him being ignorant. But the use of letters this way is something it’s easy to miss. You need first to realize that we don’t need to have a single way to represent a number. Which is implicit in learning, say, that you can write ‘7’ as the Roman numeral ‘VII’ or so, but I’m not sure that’s always clear. And realizing that you could use any symbol to write out ‘7’ if you agree that’s what the symbol means? That’s an abstraction tossed onto people who often aren’t really up for that kind of abstraction. And that we can have a symbol for “a number whose identity we don’t yet know”? Or even “a number whose identity we don’t care about”? Don’t blame someone for rearing back in confusion at this.

Friend 1: 'That algebra test was awful.' Friend 2: 'Toby just gave up and handed his paper in!' Toby: 'No, I finished. My mom said as long as I studied I didn't have to do any chores.' Friend 1: 'That'd eat up all your gaming hours!' Toby: 'Yep. Hey, did you know algebra letters stand for things?'
Greg Cravens’s The Buckets for the 8th of November, 2018. I’m sorry I can’t figure out the names of Toby’s friends here. Character lists, cartoonists, please.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th talks about vectors and scalars. And about the little ways that instructors in one subject can sabotage one another. In grad school I was witness to the mathematics department feeling quite put-upon by the engineering departments, who thought we were giving their students inadequate calculus training. Meanwhile we couldn’t figure out what they were telling students about calculus except that it was screwing up their understanding.

Funtime Activity: Ruining students forever. Teacher: 'Physics students must learn the difference between vectors and scalars is that scalars don't exist.' Student: 'What about 'amount of apples'?' Teacher: 'Huh? Oh, you're referring to 'distance in apple-space'.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th of November, 2018. Also, shouldn’t that be “displacement in apple-space”?

To a physicist, a vector is a size and a direction together. (At least until they get seriously into mathematical physics when they need a more abstract idea.) A scalar is a number. Like, a real-valued number such as ‘4’. Maybe a complex-valued number such as ‘4 + 6i’. Vectors are great because a lot of physics problems become easier when thought of in terms of directions and amounts in that direction.

A mathematician would start out with vectors and scalars like that. But then she’d move into a more abstract idea. A vector, to a mathematician, is a thing you can add to another vector and get a vector out. A scalar is something that’s not a vector but that, multiplied by a vector, gets you a vector out. This sounds circular. But by defining ‘vector’ and ‘scalar’ in how they interact with each other we get a really sweet flexibility. We can use the same reasoning — and the same proofs — for lots of things. Directions, yes. But also matrices, and continuous functions, and probabilities of events, and more. That’s a bit much to give the engineering student who’s trying to work out some problem about … I don’t know. Whatever they do over in that department. Truss bridges or electrical circuits or something.

Billy: 'On the 80s station I think I just heard my favorite song ever! It's about carrying a laser down some road. I think it's called 'Carry a laser' and it's all about lasers!' Cow: 'Actually, it's 'Kyrie Eleison'. It means 'Lord, have Mercy'. It has nothing at all to do with lasers.' Billy: 'Right, and 'Hip to b^2' has nothing to do with algebra.' Cow: 'That I don't know.'s
Mark Leiknes’s Cow and Boy rerun for the 9th of November, 2018. It first ran the 19th of March, 2012.

Mark Leiknes’s Cow and Boy for the 9th is really about misheard song lyrics, a subject that will never die now that we don’t have the space to print lyrics in the album lining anymore, or album linings. But it has a joke resonant with that of The Buckets, in supposing that algebra is just some bunch of letters mixed up with numbers. And Cow and Boy was always a strip I loved, as baffling as it might be to a casual reader. It had a staggering number of running jokes, although not in this installment.

Brad: 'I think I've got this worked out. It takes 500 half-inch hairs to make a good moustache. If I grow one hair a week, and each new hair grows 1/8 inch per month, I can grow a perfect moustache in ... ' Luann: '225 years, not bad!'
Greg Evans’s Luann Againn for the 9th of November, 2018. It first ran the 9th of November, 1990.

Greg Evans’s Luann Againn for the 9th shows Brad happy to work out arithmetic when it’s for something he’d like to know. The figure Luan gives is ridiculously high, though. If he needs 500 hairs, and one new hair grows in each week, then that’s a little under ten years’ worth of growth. Nine years and a bit over seven months to be exact. If a moustache hair needs to be a half-inch long, and it grows at 1/8th of an inch per month, then it takes four months to be sufficiently long. So in the slowest possible state it’d be nine years, eleven months. I can chalk Luann’s answer up to being snidely pessimistic about his hair growth. But his calculator seems to agree and that suggests something went wrong along the way.

Test Question: 'Mr Gray drove 55 mph to a city 80 miles away. He made two stops: one for 20 minutes, and one for 5. How long did it take Mr Gray to reach the city?' Student's answer: 'This made my head hurt, so I'm just going to say 'the whole trip'. You can't argue that.'
John Zakour and Scott Roberts’s Maria’s Day for the 9th of November, 2018. Again, character lists. I don’t know which of the characters this is except that he’s either very small or has an enormous pencil.

John Zakour and Scott Roberts’s Maria’s Day for the 9th is a story problem joke. It looks to me like a reasonable story problem, too: the distance travelled and the speed are reasonable, and give sensible numbers. The two stops add a bit of complication that doesn’t seem out of line. And the kid’s confusion is fair enough. It takes some experience to realize that the problem splits into an easy part, a hard part, and an easy part. The first easy part is how long the stops take all together. That’s 25 minutes. The hard part is realizing that if you want to know the total travel time it doesn’t matter when the stops are. You can find the total travel time by adding together the time spent stopped with the time spent driving. And the other easy part is working out how long it takes to go 80 miles if you travel at 55 miles per hour. That’s just a division. So find that and add to it the 25 minutes spent at the two stops.


The various Reading the Comics posts should all be at this link. Essays which discuss The Buckets are at this link. The incredibly many essays mentioning Saturday Morning Breakfast Cereal are at this link. Essays which mention Cow and Boy are at this link. Essays inspired in part by Luann, both the current-day and the vintage 1990 run, are at this link. The credibly many essays mentioning Maria’s Day are at this link.

And through the end of December my Fall 2018 Mathematics A-To-Z should have two new posts a week. You might like some of them.

Reading the Comics, November 7, 2018: Shorthand and Reruns Edition


There’s two types of comics for the second of last week’s review. There’s some strips that are reruns. There’s some that just use mathematics as a shorthand for something else. There’s four strips in all.

John Deering’s Strange Brew for the 6th uses mathematics as shorthand for demonstrating intelligence. There’s no making particular sense out of the symbols, of course. And I’d think it dangerous that Lucky seems to be using both capital X and lowercase x in the same formula. There’s often times one does use the capital and lowercase versions of a letter in a formula. This is usually something like “x is one element of the set X, which is all the possible candidates for some thing”. In that case, you might get the case wrong, but context would make it clear what you meant. But, yes, sometimes there’s no sensible alternative and then you have to be careful.

Person upstairs: 'I don't understand how Lucky keeps getting out.' Lucky, the dog, is downstairs, writing out plans: 'V_1 Q_q = X'^4 y / B(4) x^2'
John Deering’s Strange Brew for the 6th of November, 2018. The mathematics is all fine but have they considered they never did close the patio door?

Randy Glasbergen’s Glasbergen Cartoons for the 6th uses mathematics as shorthand for a hard subject. It’s certainly an economical notation. Alas, you don’t just learn from your mistakes. You learn from comparing your mistakes to a correct answer. And thinking about why you made the mistakes you did, and how to minimize or avoid those mistakes again.

Student, to teacher: 'If we learn from our mistakes, shouldn't I make as many mistakes as possible?' On the blackboard is the problem 473 x 17.
Randy Glasbergen’s Glasbergen Cartoons for the 6th of November, 2018. The strip is a rerun, but I have no information to date when it’s from.

So how would I do this problem? Well, carrying out the process isn’t too hard. But what do I expect the answer to be, roughly? To me, I look at this and reason: 473 is about 500. So 473 x 17 is about 500 x 17. 500 x 17 is 1000 times eight-and-a-half. So start with “about 8500”. That’s too high, obviously. I can do better. 8500 minus some correction. What correction? Well, 473 is roughly 500 minus 25. So I’ll subtract 25 times 17. Which isn’t hard, because 25 times 4 is 100. So 25 times 17? That’s 25 times 16 plus 25 times 1. 25 times 16 is 100 times 4. So 25 times 17 is 425. 8500 minus 425 is 8075. I’m still a bit high, by 2 times 17. 2 times 17 is 34. So subtract 34 from 8075: it should be about 8041.

Student taking a test: 'Which of these is a right triangle?' (Figures a, b, and c.) Maria's answer: 'c is right. a is left. b is middle.'
John Zakour and Scott Roberts’s Maria’s Day for the 7th of November, 2018. Kind of curious whether this is meant to have one correct answer or an unspecified number of correct answers, since both ‘b’ and ‘c’ look plausibly like right triangles to me.

John Zakour and Scott Roberts’s Maria’s Day for the 7th is a joke built on jargon. Every field has its jargon. Some of it will be safely original terms: people’s names (“Bessel function”) or synthetic words (“isomorphism”) that can’t be easily confused with everyday language. But some of it will be common terms given special meaning. “Right” angles and “right” triangles. “Normal” numbers. “Group”. “Right” as a description for angles and triangles goes back a long way, at least to — well, Merriam-Webster.com says 15th century. But EtymologyOnline says late 14th century. Neither offers their manuscripts. I’ll chalk it up to differences in how they interpret the texts. And possibly differences in whether they would count, say, a reference to “a right angle” written in French or German rather than in English directly.

Literary Calendar. Several jokes, including: Saturday 7pm: an infinite number of chimpanzees discuss their multi-volume 'Treasury of Western Literature with no Typos' at the Museum of Natural History. Nit picking to follow.
Richard Thompson’s Richard’s Poor Almanac rerun for the 7th of November, 2016.

Richard Thompson’s Richard’s Poor Almanac for the 7th has been run before. It references the Infinite Monkey Theorem. The monkeys this time around write up a treasury of Western Literature, not merely the canon of Shakespeare. That’s at least as impressive a feat. Also, while this is a rerun — sad to say Richard Thompson died in 2016, and was forced to retire from drawing before that — his work was fantastic and deserves attention.


This and every Reading the Comics post should be at this link. Essays discussing topics raised by Strange Brew are at this link. The essays discussing Glasbergen Cartoons are at this link. Essays which mention Maria’s Day, are at this link. And essays featuring Richard’s Poor Almanac are at this link.

My Fall 2018 Mathematics A-To-Z averages two new posts a week, through the end of December. Thanks again for reading.

Reading the Comics, November 5, 2018: November 5, 2018 Edition


This past week included one of those odd days that’s so busy I get a column’s worth of topics from a single day’s reading. And there was another strip (the Cow and Boy rerun) which I might have roped in had the rest of the week been dead. The Motley rerun might have made the cut too, for a reference to E = mc^2 .

Jason Chatfield’s Ginger Meggs for the 5th is a joke about resisting the story problem. I’m surprised by the particulars of this question. Turning an arithmetic problem into counts of some number of particular things is common enough and has a respectable history. But slices of broccoli quiche? I’m distracted by the choice, and I like quiche. It’s a weird thing for a kid to have, and a weird amount for anybody to have.

Mr Crackett: 'Alright, Meggs. Here's one for you. If Fitzcloon had 15 slices of broccoli quiche and you took a third, what would you have?' Meggs: 'A bucket ready to catch my vom---' Crackett: 'MEGGS!'
Jason Chatfield’s Ginger Meggs for the 5th of November, 2018. I’m of the age cohort to remember Real Men Don’t Eat Quiche being a book people had for some reason. Also not understanding why “real men” would not eat quiche. If you named the same dish “Cheddar Bacon Pie” you’d have men lined up for a quarter-mile to get it. Anyway, it took me too long to work out but I think the teacher’s name is Mr Crackett? Cast lists, cartoonists. We need cast lists on your comic’s About pages.

JC Duffy’s Lug Nuts for the 5th uses mathematics as a shorthand for intelligence. And it particularly uses π as shorthand for mathematics. There’s a lot of compressed concepts put into this. I shouldn’t be surprised if it’s rerun come mid-March.

The Thinking Man's Team: The Portland Pi. Shows a baseball cap with the symbol pi on it.
JC Duffy’s Lug Nuts for the 5th of November, 2018. OK, some of these strips I don’t need a cast list for.

Tom Toles’s Randolph Itch, 2 am for the 5th I’ve highlighted before. It’s the pie chart joke. It will never stop amusing me, but I suppose I should take Randolph Itch, 2 am out of my rotation of comics I read to include here.

Randolph dreaming about his presentation: pie chart. Pies have hit him and his podium, per the chart: '28% landed on stage, 13% back wall, 22% glancing blow off torso, 12% hit podium, 25% direct hit in face'. Footer joke: 'I turn now to the bar graph.'
Tom Toles’s Randolph Itch, 2 am for the 5th of November, 2018. I never get to presentations like this. It’s always someone explaining the new phone system.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 5th is a logic puzzle joke. And a set theory joke. Dad is trying to argue he can’t be surprised by his gift because it’ll belong to one of two sets of things. And he receives nothing. This ought to defy his expectations, if we think of “nothing” as being “the empty set”. The empty set is an indispensable part of set theory. It’s a set that has no elements, has nothing in it. Then suppose we talk about what it means for one set to be contained in another. Take what seems like an uncontroversial definition: set A is contained in set B if there’s nothing in A which is not also in B. Then the empty set is contained inside every set. So Dad, having supposed that he can’t be surprised, since he’d receive either something that is “socks” or something that is “not-socks”, does get surprised. He gets the one thing that is both “socks” and “not-socks” simultaneously.

Kids: 'Daddy, we got you a surprise!' Dad: 'Impossible! I assume the surprise is socks. Thus in case 1 where you get me socks, I am not surprised. In case 2, you got me not-socks. Given that I KNOW you will not give me socks because I'm anticipating socks, it's obvious the gift will be not-socks. Therefore in all cases with your gift, I remain UNSURPRISED!' Kids, after a pause: 'The gift is NOTHING!' Dad curses.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 5th of November, 2018. I may have mentioned. So my partner in Modern Physics Lab one time figured to organize his dorm room by sorting everything in it into two piles, “pair of socks” and “not a pair of socks”. I asked him how he’d classify two socks that, while mismatched, were bundled together. He informed me that he hated me.

I hate to pull this move a third time in one week (see here and here), but the logic of the joke doesn’t work for me. I’ll go along with “nothing” as being “the empty set” for these purposes. And I’ll accept that “nothing” is definitely “not-socks”. But to say that “nothing” is also “socks” is … weird, unless you are putting it in the language of set theory. I think the joke would be saved if it were more clearly established that Dad should be expecting some definite thing, so that no-thing would defy all expectations.

“Nothing” is a difficult subject to treat logically. I have been exposed a bit to the thinking of professional philosophers on the subject. Not enough that I feel I could say something non-stupid about the subject. But enough to say that yeah, they’re right, we have a really hard time describing “nothing”. The null set is better behaved. I suppose that’s because logicians have been able to tame it and give it some clearly defined properties.

Mega Lotto speaker: 'Hmm, what are the odds? First he wins the lottery and then ... ' A torn-up check and empty shoes are all that's left as a crocodile steps out of panel.
Mike Shiell’s The Wandering Melon for the 5th of November, 2018. I am curious whether this is meant to be the same lottery winner who in August got struck by lightning. It would make the torn, singed check make more direct sense. But what are the odds someone wins the lottery, gets hit by lightning, and then eaten by a crocodile? … Ah well, at least nothing worse is going to happen to him.

Mike Shiell’s The Wandering Melon for the 5th felt like a rerun to me. It wasn’t. But Shiell did do a variation on this joke in August. Both are built on the same whimsy of probability. It’s unlikely one will win a lottery. It’s unlikely one will die in a particular and bizarre way. What are the odds someone would have both things happen to them?


This and every Reading the Comics post should be at this link. Essays that include Ginger Meggs are at this link. Essays in which I discuss Lug Nuts are at this link. Essays mentioning Randolph Itch, 2 am, should be at this link. The many essays with a mention of Saturday Morning Breakfast Cereal are at this link. And essays where I’m inspired by something in The Wandering Melon should be at this link. And, what the heck, when I really discuss Cow and Boy it’s at this link. Real discussions of Motley are at this link. And my Fall 2018 Mathematics A-To-Z averages two new posts a week, now and through December. Thanks again for reading.

Reading the Comics, November 3, 2018: Arithmetic Is Hard Edition


If there is a theme to the last comic strips from the previous week, it’s that kids find arithmetic hard. That’s a title for you.

Bill Watterson’s Calvin and Hobbes for the 2nd is one of the classics, of course. Calvin’s made the mistake of supposing that mathematics is only about getting true answers. We’ll accept the merely true, if that’s what we can get. But we want interesting. Which is stuff that’s not just true but is unexpected or unforeseeable in some way. We see this when we talk about finding a “proper” answer, or subset, or divisor, or whatever. Some things are true for every question, and so, who cares?

Also, is it really true that Calvin doesn’t know any of his homework problems? It’s possible, but did he check?

Calvin: 'I can't go outside until I finish my homework. What's five plus seven?' Hobbes: 'I don't know.' Calvin: 'I don't either.' Hobbes: 'Then write 'I don't know'.' Calvin: 'Hey that's a true answer, isn't it! I can write that for ALL of these! We're done!' As Calvin races outside one parent says, 'We'd better have a look at our prodigy's homework.'
Bill Watterson’s Calvin and Hobbes rerun for the 2nd of November, 2018. It first appeared the 4th of November, 1988.

Were I grading, I would accept an “I don’t know”, at least for partial credit, in certain conditions. Those involve the student writing out what they would like to do to try to solve the problem. If the student has a fair idea of something that ought to find a correct answer, then the student’s showing some mathematical understanding. But there are times that what’s being tested is proficiency at an operation, and a blank “I don’t know” would not help much with that.

7:30 am: Todd runs into the Beauty Rejuvenation Clinic. Later, in class. Teacher: 'So if we add these denominators ...' and Todd starts snoring, with eyes open. Teacher: 'Todd! What did I tell you about getting Botox injections in your eyelids on fractions day?!'
Patrick Roberts’s Todd the Dinosaur for the 2nd of November, 2018. I admit I’m not sure Botox works on eyelids like this. But that’s for whoever writes Reading the Comics for the Medical Stuff. Also I’m not sure what you would add the denominators for, but it’s late at night when I write this. I’m probably overlooking the obvious.

Patrick Roberts’s Todd the Dinosaur for the 2nd has an arithmetic cameo. Fractions, particularly. They’re mentioned as something too dull to stay awake through. So for the joke’s purpose this could have been any subject that has an exposition-heavy segment. Fractions do have more complicated rules than adding whole numbers do. And introducing those rules can be hard. But anything where you introduce rules instead of showing what you can do with them is hard. I’m thinking here of several times people have tried to teach me board games by listing all the rules, instead of setting things up and letting me ask “what am I allowed to do now?” the first couple turns. I’m not sure how that would translate to fractions, but there might be something.

Maria as superhero declares 'With my Mighty-Maria Heat Vision I'll divide this barricade!' She gets bounced off it. 'Drat! Division is my kryptonite!' She bounced off a sheet marked 450 / 9.
John Zakour and Scott Roberts’s Maria’s Day for the 2nd of November, 2018. I do like it when a comic strip finds reasons to stretch artistically. Superhero fantasy sequences are often good pretexts.

John Zakour and Scott Roberts’s Maria’s Day for the 2nd has another of Maria’s struggles with arithmetic. It’s presented as a challenge so fierce it can defeat even superheroes. Could be any subject, really. It’s hard to beat the visual economy of having it be a division problem, though.

Hammie to his parents: 'I'm building a giant killer robot out of legos! It's going to have rockets and lasers and all kinds of stuff!' Mom: 'I wish he'd get that excited about multiplication tables.' Dad: 'Maybe math needs to be more explodey.'
Rick Kirkman and Jerry Scott’s Baby Blues for the 3rd of November, 2018. Oh, like it’s possible to finish any even slightly complicated Lego project.

Rick Kirkman and Jerry Scott’s Baby Blues for the 3rd shows a bit of youthful enthusiasm. Hammie’s parents would rather that enthusiasm be put to memorizing multiplication facts. I’m not sure this would match the fun of building stuff. But I remember finding patterns inside the multiplication table fascinating. Like how you could start from a perfect square and get the same sequence of numbers as you moved out along a diagonal. Or tracing out where the same number appeared in different rows and columns, like how just everything could multiply into 24. Might be worth playing with some.


All of my Reading the Comics posts should be at this link. Essays where I take the chance to talk about Calvin and Hobbes are at this link. Essays that include Todd the Dinosaur are at this link. Essays with a mention of Maria’s Day should be at this link. And essays with a mention of Baby Blues are at this link. Finally, and through the rest of the year, my Fall 2018 Mathematics A-To-Z should be getting two new posts a week. Thanks again for reading.

Reading the Comics, October 30, 2018: I Spot An Error Edition


The edition title says it all. Comic Strip Master Command sent me enough strips the past week for two editions and I made an unhappy discovery about one of the comics in today’s.

Dave Coverly’s Speed Bump for the 28th is your anthropomorphic-numerals joke for the week. We get to know the lowest common denominator from fractions. It’s easier to compute anything with a fraction in it if you can put everything under a common denominator. But it’s also — usually — easier to work with smaller denominators than larger ones. It’s always okay to multiply a number by 1. It may not help, but it can always be done. This has the result of multiplying both the numerator and denominator by the same number. So suppose you have something that’s written in terms of sixths, and something else written in terms of eighths. You can multiply the first thing by four-fourths, and the second thing by three-thirds. Then both fractions are in terms of 24ths and your calculation is, probably, easier.

9, talking to 2 at the bar: 'Ah, cheer up, you're not *always* the lowest common denominator.'
Dave Coverly’s Speed Bump for the 28th of October, 2018. So how many different people do you suppose are in an anthropomorphic-numerals world like this?

So this strip is the rare one where I have to say the joke doesn’t work on mathematical grounds. Coverly was mislead by the association between “lowest” and “smallest”. 2 is going to be the lowest common denominator very rarely. Everything in the problem needs to be in terms of even denominators to start with, and even that won’t guarantee it. I hate to do that, since the point of a comic strip is humor and getting any mathematics right is a bonus. But in this case, knowing the terminology shatters the joke. Coverly would have a mathematically valid joke were 9 offering the consolation “you’re not always the greatest common divisor”, the largest number that goes into a set of numbers. But nobody thinks being called the “greatest” anything ever needs consolation, so the joke would fail all but mathematics class.

Teacher: 'Why is it important for today's kids to learn algebra? Because *I* had to learn this junk in school and now it's *your* turn, that's why!'
Randy Glasbergen’s Glasbergen Cartoons for the 29th of October, 2018. It’s a rerun, as you’d tell from the copyright date and awareness that Glasbergen died in 2015. I can’t pin down from when any more than the copyright date gives you, though.

Randy Glasbergen’s Glasbergen Cartoons for the 29th is a joke of the why-learn-mathematics model. “Because we always have done this” is not a reason compelling by the rules of deductive logic. It can have great practical value. Experience can encode things which are hard to state explicitly, or to untangle from one another. And an experienced system will have workarounds for the most obvious problems, ones that a new system will not have. And any attempt at educational reform, however well-planned or meant, must answer parents’ reasonable question of why their child should be your test case.

I do sometimes see algebra attacked as being too little-useful for the class time given. I could see good cases made for spending the time on other fields of mathematics. (Probability and statistics always stands out as potentially useful; the subjects were born from things people urgently needed to know.) I’m not competent to judge those arguments and so shall not.

Baseball player missing his left arm and with a wedge cut out of his shoulder: 'I can't play today, skipper. I gave 110% yesterday.'
Carl Skanberg’s That New Carl Smell for the 29th of October, 2018. Yeah, also, all right, but suppose he only gave 100% yesterday. Wouldn’t that be all of him gone now? I’m saying there are severe logical problems with these implications is all.

Carl Skanberg’s That New Carl Smell for the 29th is a riff on jokes about giving more than 100%. Interpreting this giving-more-than-everything as running a deficit is a reasonable one. I’ve given my usual talk about “100% of what?” enough times now; I don’t need to repeat it until I think of something fresh to say.

Kid in the sandbox: 'You're on a train travelling east at 65 mph and you leave at 7:15. Another train leaves five minutes later heading west ... ' Caption: 'Mensa Schoolyard Bullies'.
Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 30th of October, 2018. I don’t know whether they use real materials to watercolor these illustrations or simply (“simply”, he calls it) digitally paint them for that effect, but it makes for a good appearance and one that I like.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 30th uses mathematics — story problems, specifically — as icons of intelligence. I can’t speak to the Mensa experience, but intellectual types trying to out-do each other? Yes, that’s a thing that happens. I mostly dodge attempts to put me to a fun mathematics puzzle. I’m embarrassed by how long it can take me to actually do one of these, when put on the spot. (I have a similar reaction to people testing my knowledge of trivia in the stuff I actually do know a ridiculous amount about.) Mostly I hope Dave Coverly doesn’t think I’m being this kid.

All of the Reading the Comics posts should be at this link. Essays that include Speed Bump are at this link. I don’t usually have a problem with it. Essays discussing Glasbergen Cartoons should be at this link. They won’t include Glasbergen’s longrunning The Better Half comic, which as far as I can find only ever appeared here the one time anyway. It’s a new tag anyway. Essays with a mention of That New Carl Smell are at this link. It’s a new tag, though, so give it some time if you want to read anything else. Essays with a mention of Mustard and Boloney are at this link. And my Fall 2018 Mathematics A-To-Z should continue for the rest of this calendar year. And it is open for requests for more of the alphabet. Thanks for reading.

Reading the Comics, October 27, 2018: Surprise Rerun Edition


While putting together the last comics from a week ago I realized there was a repeat among them. And a pretty recent repeat too. I’m supposing this is a one-off, but who can be sure? We’ll get there. I figure to cover last week’s mathematically-themed comics in posts on Wednesday and Thursday, subject to circumstances.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 26th is a joking reminder that educational texts, including in mathematics, don’t have to be boring. We can have narrative thrust and energy. It’s a good reminder.

Caption: 'I wish all educational texts were written like Epictetus wrote.' Textbook: 'What, hapless wretch? Do you suppose int(sqrt(x^2 + x)dx = (x^2 + x)^{-1/2}? And when you eat, do you carry the food to your mouth or to your eyes? Slave!
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 26th of October, 2018.

As fits the joke, the bit of calculus in this textbook paragraph is wrong. \int \sqrt{x^2 + x} dx does not equal \left(x^2 + x\right)^{-\frac12} . This is even ignoring that we should expect, with an indefinite integral like this, a constant of integration. An indefinite integral like this is equal to a family of related functions. But it’s common shorthand to write out one representative function. But the indefinite integral of \sqrt{x^2 + x} is not \left(x^2 + x\right)^{-\frac12} . You can confirm that by differentiating \left(x^2 + x\right)^{-\frac12} . The result is nothing like \sqrt{x^2 + x} . Differentiating an indefinite integral should get the original function back. Here are the rules you need to do that for yourself.

As I make it out, a correct indefinite integral would be:

\int{\sqrt{x^2 + x} dx} = \frac{1}{4}\left( \left(2x + 1\right)\sqrt{x^2 + x} + \log \left|\sqrt{x} + \sqrt{x + 1} \right| \right)

Plus that “constant of integration” the value of which we can’t tell just from the function we want to indefinitely-integrate. I admit I haven’t double-checked that I’m right in my work here. I trust someone will tell me if I’m not. I’m going to feel proud enough if I can get the LaTeX there to display.

Berle: 'It was just a happy stroll through the gloomy graveyard when suddenly ... ' Spivak's Calculus, 3rd Edition appears. Berle: 'Math jumped out of nowhere!' Harvey: 'Drink it off.'
Stephen Beals’s Adult Children rerun for the 27th of October, 2018. It also appeared the 31st of March, 2018. I’m surprised it was that recent. I can’t blame Beals if he needed a break. This is a Halloween-ready example of the comic.

Stephen Beals’s Adult Children for the 27th has run already. It turned up in late March of this year. Michael Spivak’s Calculus is a good choice for representative textbook. Calculus holds its terrors, too. Even someone who’s gotten through trigonometry can find the subject full of weird, apparently arbitrary rules. And formulas like those in the above paragraph.

Manfred: 'How do you want to divide up the bill?' Wink: 'Lemme see it .. add that and that ... carry the two ... let's just split it htree ways.' Manfred: 'Nice try.' Dusty: 'Mr Surf 'n' Turf needs a calculator!'
Rob Harrell’s Big Top rerun for the 27th of October, 2018. It originally appeared the 13th of December, 2008.

Rob Harrell’s Big Top for the 27th is a strip about the difficulties of splitting a restaurant bill. And they’ve not even got to calculating the tip. (Maybe it’s just a strip about trying to push the group to splitting the bill a way that lets you off cheap. I haven’t had to face a group bill like this in several years. My skills with it are rusty.)

Jack-o-lantern standing on a scale: 'Hey! I weigh exactly 3.14 pounds!' Caption: 'Pumpkin Pi'.
Dave Whamond’s Reality Check for the 27th of October, 2018. Does the weight count if the jack-o-lantern is wearing sneakers?

Dave Whamond’s Reality Check for the 27th is a Pi Day joke shifted to the Halloween season.


And I have more Reading the Comics post at this link. Since it’s not true that every one of these includes a Saturday Morning Breakfast Cereal mention, you can find those that have one at this link. Essays discussing Adult Children, including the first time this particular strip appeared, are at this link. Essays with a mention of Big Top are at this link. And essays with a mention of Reality Check are at this link. Furthermore, this month and the rest of this year my Fall 2018 Mathematics A-To-Z should continue. And it is open for requests for more of the alphabet.

Reading the Comics, October 26, 2018: I Am Overloaded Edition


I’ve settled to a pace of about four comics each essay. It makes for several Reading the Comics posts each week. But none of them are monsters that eat up whole evenings to prepare. Except that last week there were enough comics which made my initial cut that I either have to write a huge essay or I have to let last week’s strips spill over to Sunday. I choose that option. It’s the only way to square it with the demands of the A to Z posts, which keep creeping above a thousand words each however much I swear that this next topic is a nice quick one.

Roy Schneider’s The Humble Stumble for the 25th has some mathematics in a supporting part. It’s used to set up how strange Tommy is. Mathematics makes a good shorthand for this. It’s usually compact to write, important for word balloons. And it’s usually about things people find esoteric if not hilariously irrelevant to life. Tommy’s equation is an accurate description of what centripetal force would be needed to keep the Moon in a circular orbit at about the distance it really is. I’m not sure how to take Tommy’s doubts. If he’s just unclear about why this should be so, all right. Part of good mathematical learning can be working out the logic of some claim. If he’s not sure that Newtonian mechanics is correct — well, fair enough to wonder how we know it’s right. Spoiler: it is right. (For the problem of the Moon orbiting the Earth it’s right, at least to any reasonable precision.)

Tommy: 'Newton's theory states that the centripetal force holding the moon in its orbit must equal mv^2/R = mv^2/(60 R_E), but I'm not sure I agree.' Molly stares at him a while, and then shouts, 'TAKE THE PENCILS OUTTA YOUR NOSE!'
Roy Schneider’s The Humble Stumble rerun for the 25th of October, 2018. It originally ran the 30th of January, 2007.

Stephan Pastis’s Pearls Before Swine for the 25th shows how we can use statistics to improve our lives. At least, it shows how tracking different things can let us find correlations. These correlations might give us information about how to do things better. It’s usually a shaky plan to act on a correlation before you have a working hypothesis about why the correlation should hold. But it can give you leads to pursue.

Pig, writing out on paper: 'Percentage of my problems that occur during my waking hours: 100%. Percentage of my problems that occur when I am asleep in bed: 0%' Next panel: Pig, in bed, explaining to Rat, 'Bed is mathematically correct.'
Stephan Pastis’s Pearls Before Swine for the 25th of October, 2018. And from this we learn that Pig is not yet of the age where sometimes your neck and back hurt for three weeks because your pillow was a quarter-inch off its normal position. Bodies are fun things and everyone should have one.

Shaenon K Garrity and Jeffrey C Wells’s Skin Horse for the 25th is a joke about mathematics being hard. In this case even for a being that’s a natural mathematician. Relatable.

Virginia: 'Ms Delphi, can you calculate launch vectors?' Delphi: 'Do you think Madame Delphi's powers extend to doing MATHS?' Virginia: 'Not exactly. I think you're the analytic and predicative node of an incredibly advanced hive mind .. which, yes, does involve some math.' Delphi: 'Madame Delphi demands a pencil.'
Shaenon K Garrity and Jeffrey C Wells’s Skin Horse for the 25th of October, 2018. Uh, the characters here are trapped inside an artificial intelligence and Virginia (the black-haired woman with clear glasses) has worked out the other characters are part of an intelligent swarm of bees and they’re trying to launch an escape, so this is why all the plot makes sense.

Eric the Circle for the 26th, this one by Vissoro, is a “two types of people in the world” joke. Given the artwork I believe it’s also riffing on the binary-arithmetic version of the joke. Which is, “there are 10 types of people in the world, those who understand binary and those who don’t”.

Eric, a circle, inside a field of 1's. Caption; 'There are two types of people in the world. The ones that know about Eric, and the ones that don't.'
Eric the Circle for the 26th of October, 2018, this one by Vissoro. It makes me think of that Futurama scene where Bender has a nightmare, dreaming of a 2.

If you’d like to see more Reading the Comics posts, try this link. Essays mentioning The Humble Stumble are at this link. Essays discussing by Pearls Before Swine are at this link. Essays with a mention of Skin Horse should be at this link. I’m surprised to learn there are others, too. I’d have thought it was a new tag. Posts about what’s brought up by Eric the Circle should be at link. And this month and the rest of this year my Fall 2018 Mathematics A-To-Z should continue. And it is open for requests for more of the alphabet.

Reading the Comics, October 25, 2018: How To Save Your Tangled Earbuds Edition


The Playful Mathematics Education Blog Carnival has moved on! My successor, edition number 122, is at ArithmophobiaNoMore.com, with another mixture of the amusing, the informative, and the educational. Do please enjoy. Now on to filling out last week’s comic strips.

Brian Fies’s The Last Mechanical Monster for the 24th is a repeat. I included it last October, when I first saw it on GoComics. Still, the equations in it are right, for ballistic flight. Ballistic means that something gets an initial velocity in a particular direction and then travels without any further impulse. Just gravity. It’s a pretty good description for any system where acceleration’s done for very brief times. So, things fired from guns. Rockets, which typically have thrust for a tiny slice of their whole journey and coast the rest of the time. Anything that gets dropped. Or, as in here, a mad scientist training his robot to smash his way through a bank, and getting flung so.

Mad Scientist, tossed in the air by a rampaging robot: 'I realized my error the moment I uttered it. A ballistic arc is described by d = v cos(theta)/g (v sin(theta) + sqrt((v sin(theta))^2 + 2 g y_0)). Estimating initial velocity and angle and solving for t = d/(v cos(theta)) I'll hit the ground in about one and one-quarter seconds.
Brian Fies’s The Last Mechanical Monster rerun for the 24th of October, 2018. It’s also appeared the 4th of October, 2017.

The symbols in the equations are not officially standardized. But they might as well be. ‘v’ here means the speed that something’s tossed into the air. It really wants to be ‘velocity’, but velocity, in the trades, carries with it directional information. And here that’s buried in ‘θ’, the angle with respect to vertical that the thing starts flight in. ‘g’ is the acceleration of gravity, near enough constant if you don’t travel any great distance over the surface of the Earth. ‘y0‘ is the height from which the thing started to fly. And so then ‘d’ becomes the distance travelled, while ‘t’ is the time it takes to travel. I’m impressed the mad scientist (the one from the original Superman cartoon, in 1941; Fies wrote a graphic novel about that man after his release from jail in the present day.)

2, speaking to a bouncer and pointing to the '1 +' to its (stage) left: 'Name's Deuce, I'm on the list. This is my plus-one.'
Dan Thompson’s Brevity for the 24th of October, 2018. I understand why sensibly the ‘2’ would be the character nearest the bouncer. But wouldn’t the joke read better if ‘2’ were on the left, so the panel read 2 + 1?

Dan Thompson’s Brevity for the 24th is the anthropomorphic numerals joke for this essay.

Greg Cravens’s Hubris! for the 24th jokes about the dangers of tangled earbuds. For once, mathematics can help! There’s even a whole field of mathematics about this. Not earbuds specifically, but about knots. It’s called knot theory. I trust field was named by someone caught by surprise by the question. A knot, in this context, is made of a loop of thread that’s assumed to be infinitely elastic, so you can always stretch it out or twist it around some. And it’s frictionless, so you can slide the surface against itself without resistance. And you can push it along an end. These are properties that real-world materials rarely have.

Alien: 'Listen, I have this ultra high wave emitter I've been using to hobble the other teams but it's quit working.' Computer Repair Guy: 'let me see ... hang on, not a software issue, ah, hey, here's your issue. All fixed! $50.' Alien: 'Great! Hey, I also have these ear buds that tangle up so badly that ... ' Repair Guy: 'Yi-i-i-kes. Nothing to be done. Toss 'em.'
Greg Cravens’s Hubris! for the 24th of October, 2018. So this is all happening in the midst of a sports festival, which is why there’s a Grey alien in Charlie Brown outfit. It’s a bit of a weird comic, but I like it.

But. They can be close enough. And knot theory tells us some great, useful stuff. Among them: your earbuds are never truly knotted. To be a knot at all, the string has to loop back and join itself. That is, it has to be like a rubber band, or maybe an infinity scarf. If it’s got loose ends, it’s no knot. It’s topologically just a straight thread with some twists made in the surface. They can come loose.

All that holds these earbuds together is the friction of the wires against each other. (That the earbud wire splits into a left and a right bud doesn’t matter, here.) They can be loosened. Let me share how.

My love owns, among other things, a marionette dragon. And once, despite it being stored properly, the threads for it got tangled, and those things are impossible to untangle on purpose. I, having had one (1) whole semester of knot theory in grad school, knew an answer. I held the marionette upside-down, by the dragon. The tangled wires and the crossed sticks that control it hung loose underneath. And then shook the puppet around. This made the wires, and the sticks, shake around. They untangled, quickly.

What held the marionette strings, and what holds earbuds, together, is just friction. It’s hard to make the wire slide loosely against itself. Shaking it around, though? That gives it some energy. That gives the wire some play. And here we have one of the handful of cases where entropy does something useful for us. There’s a limit to how tightly a wire can loop around itself. There’s no limit to how loosely it can go. Little, regular, random shakes will tend to loosen the wire. When it’s loose enough, it untangles naturally.

You can help this along. We all know how. Use a pen-point or a toothpick a needle to pry some of the wires apart. That makes the “knot” easier to remove. This works by the same principle. If you reduce how much the wire contacts itself, you reduce the friction on the wire. The wire can slide more easily into the un-knot that it truly is. The comic’s tech support guy gave up too easily.

Horace counting sheep in bed: MMCMXCVIII. MMCMXCIX. MMM (it's a cake rather than a sheep.)
Samson’s Dark Side of the Horse for the 25th of October, 2018. Better than working on Collatz conjectures in your head, anyway. 27 is a heck of a number to start from.

Samson’s Dark Side of the Horse for the 25th is the Roman numerals joke for this essay. And a cute bit about coincidences between what you can spell out with Roman numerals and sounds people might make. Writing out calculations evokes peculiar, magical prowess. When they include, however obliquely, words? Or parts of words? Can’t blame people for seeing the supernatural in it.


I can’t promise that every one of these Reading the Comics posts will be able to solve your minor problems. But if you want to try, you can read them here. The other essays mentioning The Last Mechanical Monster are at this link. Essays discussing ideas brought up by Brevity are at this link. Essays discussing Hubris will be at this link. It’s a new tag, though, so there’s only this post on it right now. Posts featuring Dark Side Of The Horse should be at link. And I do continue posting my Fall 2018 Mathematics A-To-Z, which is open for requests for more of the alphabet this week. Thanks for reading and thanks for making suggestions.

Reading the Comics, October 24, 2018: Frazz Really Wants To Be My Friend Edition


It’s another week with several on-topic installments of Frazz. Again, Jef Mallet, you and I live in the same metro area. Wave to me at the farmer’s market or something. I’m kind of able to talk to people in real life, if I can keep in view three different paths to escape and know two bathrooms to hide in. Horrock’s is great for that.

Jef Mallet’s Frazz for the 22nd is a bit of wordplay. It’s built on the association between “negative” and “wrong”. And the confusing fact that multiplying a negative number by a negative number results in a positive number. It sounds like a trick. Still, negative numbers are tricky. The name connotes something that’s gone a bit wrong. It took time to understand what they were and how they should work. This weird multiplication rule follows from that. If we don’t suppose this to be true, then we break other ideas we have about multiplication and comparative sizes and such. Mathematicians needed to get comfortable with negative numbers. For a long time, for example, mathematicians would treat x^2 - 4x + 4 = 0 and x^4 + 4x + 4 = 0 as different kinds of polynomials to solve. Today we see a -4 as no harder than a +4, now that we’re good at multiplying it out. And I have read, but have not seen explained, that there was uncertainty among the philosophers of mathematics about whether we should consider negative numbers, as a group, to be greater than or less than positive numbers. (I have reasons for thinking this a mighty interesting speculation.) There’s reasons to doubt them, is what I have to say.

Mrs Olsen: 'Any questions? Goody. Caulfield.' Caulfield: 'If a negative times a negative is a positive, how come two wrongs don't make a right?' [Later] Frazz: 'Maybe negative isn't the same as wrong.' Caulfield: 'You are not incorrect.'
Jef Mallet’s Frazz for the 22nd of October, 2018. Good thing to learn, really.

Bob Weber Jr and Jay Stephens’s Oh Brother for the 22nd reminds me of my childhood. At some point I was pairing up the counting numbers and the letters of the alphabet, and realized that the alphabet ended while the numbers did not. Something about that offended my young sense of justice. I’m not sure how, anymore. But that it was always possible to find a bigger number than whatever you thought was the biggest caught my imagination.

Bud: 'Lily! Lily! What's the biggest number?' Lily: 'It's the same as the number of times you bug me.' Bud: 'But that's an ongoing, never-ending number.' Lily: 'Exactly!' Bud: 'Thanks for explaining math in practical terms!'
Bob Weber Jr and Jay Stephens’s Oh Brother for the 22nd of October, 2018. This may be a rerun; I don’t know if the strip is still in original production.

There is, surely, a largest finite number that anybody will ever use for something, even if it’s just hyperbole. I’m curious what it will be. Surely we can’t have already used it. A number named Skewes’s Number was famous, for a while, as the largest number actually used in a proof of something. The fame came from Isaac Asimov writing an essay about the number, and why someone might care, and how hard it is just describing how big the number is in a comprehensible way. Wikipedia tells me this number’s far been exceeded by, among other things, something called Rayo’s Number. It’s “the smallest number bigger than any finite number named by an expression in the language of set theory with a googol symbols or less” (plus some technical points to keep you from cheating). Which, all right, but I’d like to know if we think the first digit is a 1, maybe a 2? Somehow I don’t demand that of Skewes, perhaps because I read that Asimov essay when I was at an impressionable age.

Caulfield: 'If a fraction divided by a fraction is just a fraction times a flipped fraction, what happens if you fish with a fly for flying fish?' Mrs Olsen: 'You can't wade that far out in the ocean.' [ Later ] Frazz: 'So, nothing.' Caulfield: 'I don't think you can divide a fraction by a fraction and get zero.'
Jef Mallet’s Frazz for the 23rd of October, 2018. I appreciate when Mrs Olsen is given the chance to show she does know things.

Jef Mallet’s Frazz for the 23rd has Caulfield talk about a fraction divided by a fraction. And particularly he says “a fraction divided by a fraction is just a fraction times a flipped fraction”. This offends me, somehow. This even though that is how I’d calculate the value of the division, if I needed to know that. But it seems to me like automatically going to that process skips recognizing that, say, \frac{2}{5} \div \frac{1}{10} shouldn’t be surprising if it turns out not to be a fraction. Well, Caulfield’s just looking to cause trouble with a string of wordplay. I can think of how to divide a fraction by a fraction and get zero.

One is really the only number there is! All other numbers are simply collections of ones.
Ashleigh Brilliant’s Pot-Shots for the 23rd of October, 2018. This is a rerun, but from 1977; the strip is not in regular production anymore.

Ashleigh Brilliant’s Pot-Shots for the 23rd promises to recapitulate the whole history of mathematics in a single panel. Ambitious bit of work. It’s easy to picture going from the idea of 1 to any of the positive whole numbers, though. It’s so easy it doesn’t even need humans to do it; animals can count, at least a bit. We just carry on to a greater extent than the crows or the raccoons do, so far as we’ve heard. From those, it takes some squinting, but you can think of negative whole numbers. And from that you get zero pretty quickly. You can also get rational numbers. The western mathematical tradition did this by looking at … er … ratios, that something might be to another thing as two is to five. Circumlocutions like that. Getting to irrational numbers is harder. Can be harder. Some irrational numbers beg you to notice them: the square root of two, for example. Square root of three. Numbers that come up from solving polynomial equations. But there are more number than those. Many more numbers. You might suspect the existence of a transcendental number, that isn’t the root of any polynomial that’s decently behaved. But finding one? Or finding that there are more transcendental number than there are real numbers? This takes a certain brilliance to suspect, and to prove out. But we can get there with rational numbers — which we get to from collections of ones — and the idea of cutting sets of numbers into those smaller than and those bigger than something. Ashleigh Brilliant has more truth than, perhaps, he realized when he drew this panel.

Goldfish, in a tank, to its peers: 'This may seem weird, but my research indicates that the universe has the shape of a perfect cuboid.
Niklas Eriksson’s Carpe Diem for the 24th of October, 2018. I’m curious how the ground is accounted for.

Niklas Eriksson’s Carpe Diem for the 24th has goldfish work out the shape of space. A goldfish in this case has the advantage of being able to go nearly everywhere in the space. But working out what the universe must look like, when you can only run local experiments, is a great geometric problem. It’s akin to working out that the Earth must be a sphere, and about how big a sphere, from the surveying job one can do without travelling more than a few hundred kilometers.


If you’re interested in reading the comics, you might want to see Reading the Comics posts. They’re here. More essays mentioning Frazz should be at this link. Essays that discuss ideas brought up by Oh Brother! should be this link. Essays which talk about Frazz — wait. I said that. This and other appearances by Pot Shots should be at this link. And posts which feature Carpe Diem should be at link. Do please stick around for more of my Fall 2018 Mathematics A-To-Z, too. I’m trying to keep up at two essays a week through the end of the year, which is not precisely fall.

Reading the Comics, October 19, 2018: More Short Things Edition


At least, I’d thought the last half of last week’s comics were mostly things I could discuss quickly. Then Frank and Ernest went and sprawled on me. Such will happen.

Before I get to that, I did want to mention that Gregory Taylor’s paneling for votes for the direction his mathematics-inspired serial takes:

You may enjoy; at least, give it a try.

Thaves’s Frank and Ernest for the 18th is a bit of wordplay. There’s something interesting culturally about phrasing “lots of math, but no chemistry”. Algorithms as mathematics makes sense. Much of mathematics is about finding processes to do interesting things. Algorithms, and the mathematics which justifies them, can at least in principle be justified with deductive logic. And we like to think that the universe must make deductive-logical sense. So it is easy to suppose that something mathematical simply must make logical sense.

Frank: 'It didn't go well? That date was selected for you by a sophisticated statistical algorithm.' Ernest: 'Lots of math, but no chemistry.'
Thaves’s Frank and Ernest for the 18th of October, 2018. One might argue this just represents the algorithm not having enough data. That there are aspects to both people which were poorly modelled. Could happen.

Chemistry, though. It’s a metaphor for whatever the difference is between a thing’s roster of components and the effect of the whole. The suggestion is that it is mysterious and unpredictable. It’s an attitude strange to actual chemists, who have a rather good understanding of why most things happen. My suspicion is that this sense of chemistry is old, dating to before we had a good understanding of why chemical bonds work. We have that understanding thanks to quantum mechanics, and its mathematical representations.

But we can still allow for things that happen but aren’t obvious. When we write about “emergent properties” we describe things which are inherent in whatever we talk about. But they only appear when the things are a large enough mass, or interact long enough. Some things become significant only when they have enough chance to be seen.

Zeno: 'Honey, I'd love to, but it's not as if I can traverse infinite regions in finite time!' Caption: 'Fun Fact: Zeno never took out the garbage.'
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th of October, 2018. I wrote all this text assuming Weinersmith meant Zeno of Elea. It’d be a heck of a thing if he meant Zeno, the Omni-King of the 12 Universes that I’m told is a thing in Dragon Ball. I guess they have different character designs.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th is about mathematicians’ favorite Ancient Greek philosopher they haven’t actually read. (In fairness, Zeno is hard to read, even for those who know the language.) Zeno’s famous for four paradoxes, the most familiar of which is alluded to here. To travel across a space requires travelling across half of it first. But this applies recursively. To travel any distance requires accomplishing infinitely many partial-crossings. How can you do infinitely many things, each of which take more than zero time, in less than an infinitely great time? But we know we do this; so, what aren’t we understanding? A callow young mathematics major would answer: well, pick any tiny interval of time you like. All but a handful of the partial-crossings take less than your tiny interval time. This seems like a sufficient answer and reason to chuckle at philosophers. Fine; an instant has zero time elapse during it. Nothing must move during that instant, then. So when does movement happen, if there is no movement during all the moments of time? Reconciling these two points slows the mathematician down.

Teacher: 'Todd, please come up to the chalkboard and do this fraction problem.' Todd: 'Uh-oh! It's late October, almost November! And hibernation is kicking in! I'll have the answer for ya next spring! Well, see ya!' Teacher: 'Todd! Get up here right now!' Student: 'Teacher, it's not safe to wake a hibernating T-Rex! Especially when I'm the first one he'll see!'
Patrick Roberts’s Todd the Dinosaur for the 19th of October, 2018. I’d imagine he would have a sufficient excuse in his arms not being able to reach the chalkboard, actually.

Patrick Roberts’s Todd the Dinosaur for the 19th mentions fractions. It’s only used to list a kind of mathematics problem a student might feign unconsciousness rather than do. And takes quite little space in the word balloon to describe. It’d be the same joke if Todd were asked to come up and give a ten-minute presentation on the Battle of Bunker Hill.

Penny: 'It says here the Rubik's Cube is still a top-selling gift for kids at Christmas.' Earl: 'Why for kids?? *I* can't even do it!' Burl: 'But it's great for kids. It keeps Timmy busy for hours, even though the poor kid doesn't realize yet that it's impossible to do.
Julie Larson’s The Dinette Set for the 19th of October, 2018. It ran originally the 12th of December, 2007. Among the stray, side, filler jokes, I like best that both Earl and Burl have mugs reading “And Dumber”.

Julie Larson’s The Dinette Set for the 19th mentions the Rubik’s Cube. Sometime I should do a proper essay about its mathematics. Any Rubik’s Cube can be solved in at most 20 moves. And it’s apparently known there are some cube configurations that take at least 20 moves, so, that’s nice to have worked out. But there are many approaches to solving a cube, none of which I am competent to do. Some algorithms are, apparently, easier for people to learn, at the cost of taking more steps. And that’s fine. You should understand something before you try to do it efficiently.

Venn diagram of Content Warning. Bubble A: Violence. Bubble B: Mature Subject Matter. Bubble C: Coarse Language. Intersection of A and B: The Bible. Intersection of B and C: Thanksgiving Dinner. Intersection of C and A: Sports. Intersection of A, B, and C: Life.
John Atkinson’s Wrong Hands for the 19th of October, 2018. So … life is the intersection of Thanksgiving, the Bible, and Sports? … I could see the case for that.

John Atkinson’s Wrong Hands for the 19th is the Venn Diagram joke for the week. Good to have one around.


This and my other Reading the Comics posts are available at this link. The essays mentioning Frank and Ernest should be at this link. For just the Reading the Comics posts with Saturday Morning Breakfast Cereal content try this link. Essays which talk about things raised by Todd the Dinosaur are at this link. Posts that write about The Dinette Set are at this link. And the essays based on Wrong Hands should be at this link. And do please stick around for more of my Fall 2018 Mathematics A-To-Z, with another post due tomorrow that I need to write today.

My 2018 Mathematics A To Z: Infinite Monkey Theorem


Dina Yagodich gave me the topic for today. She keeps up a YouTube channel with a variety of interesting videos. And she did me a favor. I’ve been thinking a long while to write a major post about this theorem. Its subject turns up so often. I’d wanted to have a good essay about it. I hope this might be one.

Cartoon of a thinking coati (it's a raccoon-like animal from Latin America); beside him are spelled out on Scrabble titles, 'MATHEMATICS A TO Z', on a starry background. Various arithmetic symbols are constellations in the background.
Art by Thomas K Dye, creator of the web comics Newshounds, Something Happens, and Infinity Refugees. His current project is Projection Edge. And you can get Projection Edge six months ahead of public publication by subscribing to his Patreon. And he’s on Twitter as @Newshoundscomic.

Infinite Monkey Theorem.

Some mathematics escapes mathematicians and joins culture. This is one such. The monkeys are part of why. They’re funny and intelligent and sad and stupid and deft and clumsy, and they can sit at a keyboard almost look in place. They’re so like humans, except that we empathize with them. To imagine lots of monkeys, and putting them to some silly task, is compelling.

Monkey Typewriter Theory: An immortal monkey pounding on a typewriter will eventually reproduce the text of 'Hamlet'. Baby Keyboard Theory: Left alone, a baby pounding on a computer keyboard will eventually order 32 cases of bathroom caulk from an online retailer.
Paul Trapp’s Thatababy for the 13th of February, 2014.

The metaphor traces back to a 1913 article by the mathematical physicist Émile Borel which I have not read. Searching the web I find much more comment about it than I find links to a translation of the text. And only one copy of the original, in French. And that page wants €10 for it. So I can tell you what everybody says was in Borel’s original text, but can’t verify it. The paper’s title is “Statistical Mechanics and Irreversibility”. From this I surmise that Borel discussed one of the great paradoxes of statistical mechanics. If we open a bottle of one gas in an airtight room, it disperses through the room. Why doesn’t every molecule of gas just happen, by chance, to end up back where it started? It does seem that if we waited long enough, it should. It’s unlikely it would happen on any one day, but give it enough days …

But let me turn to many web sites that are surely not all copying Wikipedia on this. Borel asked us to imagine a million monkeys typing ten hours a day. He posited it was possible but extremely unlikely that they would exactly replicate all the books of the richest libraries of the world. But that would be more likely than the atmosphere in a room un-mixing like that. Fair enough, but we’re not listening anymore. We’re thinking of monkeys. Borel’s is a fantastic image. It would see some adaptation in the years. Physicist Arthur Eddington, in 1928, made it an army of monkeys, with their goal being the writing all the books in the British Museum. By 1960 Bob Newhart had an infinite number of monkeys and typewriters, and a goal of all the great books. Stating the premise gets a laugh I doubt the setup would today. I’m curious whether Newhart brought the idea to the mass audience. (Google NGrams for “monkeys at typewriters” suggest that phrase was unwritten, in books, before about 1965.) We may owe Bob Newhart thanks for a lot of monkeys-at-typewriters jokes.

Kid: 'Mom, Dad, I want to go bungee jumping this summer!' Dad: 'A thousand monkeys working a thousand typewriters would have a better chance of randomly typing the complete works of William Shakespeare over the summer than you have of bungee jumping.' (Awksard pause.) Kid: 'What's a typewriter?' Dad: 'A thousand monkeys randomly TEXTING!'
Bill Hinds’s Cleats rerun for the 1st of July, 2018.

Newhart has a monkey hit on a line from Hamlet. I don’t know if it was Newhart that set the monkeys after Shakespeare particularly, rather than some other great work of writing. Shakespeare does seem to be the most common goal now. Sometimes the number of monkeys diminishes, to a thousand or even to one. Some people move the monkeys off of typewriters and onto computers. Some take the cowardly measure of putting the monkeys at “keyboards”. The word is ambiguous enough to allow for typewriters, computers, and maybe a Megenthaler Linotype. The monkeys now work 24 hours a day. This will be a comment someday about how bad we allowed pre-revolutionary capitalism to get.

The cultural legacy of monkeys-at-keyboards might well itself be infinite. It turns up in comic strips every few weeks at least. Television shows, usually writing for a comic beat, mention it. Computer nerds doing humor can’t resist the idea. Here’s a video of a 1979 Apple ][ program titled THE INFINITE NO. OF MONKEYS, which used this idea to show programming tricks. And it’s a great philosophical test case. If a random process puts together a play we find interesting, has it created art? No deliberate process creates a sunset, but we can find in it beauty and meaning. Why not words? There’s likely a book to write about the infinite monkeys in pop culture. Though the quotations of original materials would start to blend together.

But the big question. Have the monkeys got a chance? In a break from every probability question ever, the answer is: it depends on what the question precisely is. Occasional real-world experiments-cum-art-projects suggest that actual monkeys are worse typists than you’d think. They do more of bashing the keys with a stone before urinating on it, a reminder of how slight is the difference between humans and our fellow primates. So we turn to abstract monkeys who behave more predictably, and run experiments that need no ethical oversight.

Toby: 'So this English writer is like a genius, right? And he's the greatest playwright ever. And I want to be just like him! Cause what he does, see, is he gets infinite monkeys on typewriters and just lets 'em go nuts, so eventually they write ALL of Shakespeare's plays!' Brother: 'Cool! And what kind of monkey is an 'infinite'?' Toby: 'Beats me, but I hope I don't have to buy many of them.' Dad: 'Toby, are you *sure* ywou completely pay attention when your teachers are talking?' Toby: 'What? Yes! Why?'

Greg Cravens’ The Buckets for the 30th of March, 2014.

So we must think what we mean by Shakespeare’s Plays. Arguably the play is a specific performance of actors in a set venue doing things. This is a bit much to expect of even a skilled abstract monkey. So let us switch to the book of a play. This has a more clear representation. It’s a string of characters. Mostly letters, some punctuation. Good chance there’s numerals in there. It’s probably a lot of characters. So the text to match is some specific, long string of characters in a particular order.

And what do we mean by a monkey at the keyboard? Well, we mean some process that picks characters randomly from the allowed set. When I see something is picked “randomly” I want to know what the distribution rule is. Like, are Q’s exactly as probable as E’s? As &’s? As %’s? How likely it is a particular string will get typed is easiest to answer if we suppose a “uniform” distribution. This means that every character is equally likely. We can quibble about capital and lowercase letters. My sense is most people frame the problem supposing case-insensitivity. That the monkey is doing fine to type “whaT beArD weRe i BEsT tO pLAy It iN?”. Or we could set the monkey at an old typesetter’s station, with separate keys for capital and lowercase letters. Some will even forgive the monkeys punctuating terribly. Make your choices. It affects the numbers, but not the point.

Literary Calendar. Several jokes, including: Saturday 7pm: an infinite number of chimpanzees discuss their multi-volume 'Treasury of Western Literature with no Typos' at the Museum of Natural History. Nit picking to follow.
Richard Thompson’s Richard’s Poor Almanac rerun for the 7th of November, 2016.

I’ll suppose there are 91 characters to pick from, as a Linotype keyboard had. So the monkey has capitals and lowercase and common punctuation to get right. Let your monkey pick one character. What is the chance it hit the first character of one of Shakespeare’s plays? Well, the chance is 1 in 91 that you’ve hit the first character of one specific play. There’s several dozen plays your monkey might be typing, though. I bet some of them even start with the same character, so giving an exact answer is tedious. If all we want monkey-typed Shakespeare plays, we’re being fussy if we want The Tempest typed up first and Cymbeline last. If we want a more tractable problem, it’s easier to insist on a set order.

So suppose we do have a set order. Then there’s a one-in-91 chance the first character matches the first character of the desired text. A one-in-91 chance the second character typed matches the second character of the desired text. A one-in-91 chance the third character typed matches the third character of the desired text. And so on, for the whole length of the play’s text. Getting one character right doesn’t make it more or less likely the next one is right. So the chance of getting a whole play correct is \frac{1}{91} raised to the power of however many characters are in the first script. Call it 800,000 for argument’s sake. More characters, if you put two spaces between sentences. The prospects of getting this all correct is … dismal.

I mean, there’s some cause for hope. Spelling was much less fixed in Shakespeare’s time. There are acceptable variations for many of his words. It’d be silly to rule out a possible script that (say) wrote “look’d” or “look’t”, rather than “looked”. Still, that’s a slender thread.

Proverb Busters: testing the validity of old sayings. Doctor: 'A hundred monkeys at a hundred typewriters. Over time, will one of them eventually write a Shakepeare play?' Winky: 'Nope. Just the script for Grown-Ups 3'. Doctor: 'Another proverb busted.'
Tim Rickard’s Brewster Rockit for the 1st of April, 2014.

But there is more reason to hope. Chances are the first monkey will botch the first character. But what if they get the first character of the text right on the second character struck? Or on the third character struck? It’s all right if there’s some garbage before the text comes up. Many writers have trouble starting and build from a first paragraph meant to be thrown away. After every wrong letter is a new chance to type the perfect thing, reassurance for us all.

Since the monkey does type, hypothetically, forever … well, so each character has a probability of only \left(\frac{1}{91}\right)^{800,000} (or whatever) of starting the lucky sequence. The monkey will have 91^{800,000} chances to start. More chances than that.

And we don’t have only one monkey. We have a thousand monkeys. At least. A million monkeys. Maybe infinitely many monkeys. Each one, we trust, is working independently, owing to the monkeys’ strong sense of academic integrity. There are 91^{800,000} monkeys working on the project. And more than that. Each one takes their chance.

Melvin: 'Hold on now --- replacement? Who could you find to do all the tasks only Melvin can perform?' Rita: 'A macaque, in fact. Listen, if an infinite number of monkeys can write all the great works, I'm confident that one will more than cover for you.'
John Zakour and Scott Roberts’s Working Daze for the 29th of May, 2018.

There are dizzying possibilities here. There’s the chance some monkey will get it all exactly right first time out. More. Think of a row of monkeys. What’s the chance the first thing the first monkey in the row types is the first character of the play? What’s the chance the first thing the second monkey in the row types is the second character of the play? The chance the first thing the third monkey in the row types is the third character in the play? What’s the chance a long enough row of monkeys happen to hit the right buttons so the whole play appears in one massive simultaneous stroke of the keys? Not any worse than the chance your one monkey will type this all out. Monkeys at keyboards are ergodic. It’s as good to have a few monkeys working a long while as to have many monkeys working a short while. The Mythical Man-Month is, for this project, mistaken.

That solves it then, doesn’t it? A monkey, or a team of monkeys, has a nonzero probability of typing out all Shakespeare’s plays. Or the works of Dickens. Or of Jorge Luis Borges. Whatever you like. Given infinitely many chances at it, they will, someday, succeed.

Except.

A thousand monkeys at a thousand typewriters ... will eventually write 'Hamlet'. A thousand cats at a thousand typewriters ... will tell you go to write your own danged 'Hamlet'.
Doug Savage’s Savage Chickens for the 14th of August, 2018.

What is the chance that the monkeys screw up? They get the works of Shakespeare just right, but for a flaw. The monkeys’ Midsummer Night’s Dream insists on having the fearsome lion played by “Smaug the joiner” instead. This would send the play-within-the-play in novel directions. The result, though interesting, would not be Shakespeare. There’s a nonzero chance they’ll write the play that way. And so, given infinitely many chances, they will.

What’s the chance that they always will? That they just miss every single chance to write “Snug”. It comes out “Smaug” every time?

Eddie: 'You know the old saying about putting an infinite number of monkeys at an infinite number of typewriters, and eventually they'll accidentally write Shakespeare's plays?' Toby: 'I guess.' Eddie: 'My English teacher says that nothing about our class should worry those monkeys ONE BIT!'
Greg Cravens’s The Buckets for the 6th of October, 2018.

We can say. Call the probability that they make this Snug-to-Smaug typo any given time p . That’s a number from 0 to 1. 0 corresponds to not making this mistake; 1 to certainly making it. The chance they get it right is 1 - p . The chance they make this mistake twice is smaller than p . The chance that they get it right at least once in two tries is closer to 1 than 1 - p is. The chance that, given three tries, they make the mistake every time is even smaller still. The chance that they get it right at least once is even closer to 1.

You see where this is going. Every extra try makes the chance they got it wrong every time smaller. Every extra try makes the chance they get it right at least once bigger. And now we can let some analysis come into play.

So give me a positive number. I don’t know your number, so I’ll call it ε. It’s how unlikely you want something to be before you say it won’t happen. Whatever your ε was, I can give you a number M . If the monkeys have taken more than M tries, the chance they get it wrong every single time is smaller than your ε. The chance they get it right at least once is bigger than 1 – ε. Let the monkeys have infinitely many tries. The chance the monkey gets it wrong every single time is smaller than any positive number. So the chance the monkey gets it wrong every single time is zero. It … can’t happen, right? The chance they get it right at least once is closer to 1 than to any other number. So it must be 1. So it must be certain. Right?

Poncho, the dog, looking over his owner's laptop: 'They say if you let an infinite number of cats walk on an infinite number of keyboards, they'll eventually type all the great works of Shakespeare.' The cat walks across the laptop, connecting to their owner's bank site and entering the correct password. Poncho: 'I'll take it.'
Paul Gilligan’s Pooch Cafe for the 17th of September, 2018.

But let me give you this. Detach a monkey from typewriter duty. This one has a coin to toss. It tosses fairly, with the coin having a 50% chance of coming up tails and 50% chance of coming up heads each time. The monkey tosses the coin infinitely many times. What is the chance the coin comes up tails every single one of these infinitely many times? The chance is zero, obviously. At least you can show the chance is smaller than any positive number. So, zero.

Yet … what power enforces that? What forces the monkey to eventually have a coin come up heads? It’s … nothing. Each toss is a fair toss. Each toss is independent of its predecessors. But there is no force that causes the monkey, after a hundred million billion trillion tosses of “tails”, to then toss “heads”. It’s the gambler’s fallacy to think there is one. The hundred million billion trillionth-plus-one toss is as likely to come up tails as the first toss is. It’s impossible that the monkey should toss tails infinitely many times. But there’s no reason it can’t happen. It’s also impossible that the monkeys still on the typewriters should get Shakespeare wrong every single time. But there’s no reason that can’t happen.

It’s unsettling. Well, probability is unsettling. If you don’t find it disturbing you haven’t thought long enough about it. Infinities, too, are unsettling so.

Researcher overseeing a room of monkeys: 'Shakespeare would be OK, but I'd prefer they come up with a good research grant proposal.'
John Deering’s Strange Brew for the 20th of February, 2014.

Formally, mathematicians interpret this — if not explain it — by saying the set of things that can happen is a “probability space”. The likelihood of something happening is what fraction of the probability space matches something happening. (I’m skipping a lot of background to say something that simple. Do not use this at your thesis defense without that background.) This sort of “impossible” event has “measure zero”. So its probability of happening is zero. Measure turns up in analysis, in understanding how calculus works. It complicates a bunch of otherwise-obvious ideas about continuity and stuff. It turns out to apply to probability questions too. Imagine the space of all the things that could possibly happen as being the real number line. Pick one number from that number line. What is the chance you have picked exactly the number -24.11390550338228506633488? I’ll go ahead and say you didn’t. It’s not that you couldn’t. It’s not impossible. It’s just that the chance that this happened, out of the infinity of possible outcomes, is zero.

The infinite monkeys give us this strange set of affairs. Some things have a probability of zero of happening, which does not rule out that they can. Some things have a probability of one of happening, which does not mean they must. I do not know what conclusion Borel ultimately drew about the reversibility problem. I expect his opinion to be that we have a clear answer, and unsettlingly great room for that answer to be incomplete.


This and other Fall 2018 Mathematics A-To-Z posts can be read at this link. The next essay should come Friday and will, I hope, be shorter.

Reading the Comics, October 18, 2018: Quick Half-Week Edition


There were enough mathematically-themed comic strips last week to split across two essays. The first half of them don’t take too much time to explain. Let me show you.

Henry Scarpelli and Craig Boldman’s Archie for the 15th is the pie-chart wordplay joke for the week. I don’t remember there ever being pie at the high school cafeteria, but back when I was in high school I often skipped lunch to hang out in the computer room.

Jughead: 'Ummm! Nummm!' Archie: 'Quiet, Jug! We've got to get this group project finished!' Jughead: 'Mmm! I'm hungry! I'm off to the lunchroom for a snack!' Archie: 'I told you it was a mistake to include a pie chart in our report!'
Henry Scarpelli and Craig Boldman’s Archie rerun for the 15th of October, 2018. Oh gads. If Jughead makes this much noise just imagining food then when he really eats he’s got to be one of those people you can hear from the next state over. I have no information about when this strip first ran.

Will Henry’s Wallace the Brave for the 15th alludes to a report on trapezoids. I can’t imagine what about this would be so gold-star-worthy when I’ve surely already written plenty about trapezoids. … Really, that thing trying to classify how many different kinds of trapezoids there are would be my legacy to history if I hadn’t also written about how many grooves are on a record’s side.

Teacher: 'Wallace, Spud, fantastic report on trapezoids. Gold stars for each.' (Both are delighted; girl in the back says 'Lamesville.') Spud: 'I haven't gotten a gold star since I got my head stuck in that bannister.' Wallace: 'They buttered you up like an ear of corn.'
Will Henry’s Wallace the Brave for the 15th of October, 2018. All right, the strip is only marginally on topic. It and Breaking Cat News are the syndicated comic strips I’ve been most excited for since Richard Thompson wasn’t able to continue Cul de Sac.

Thaves’s Frank and Ernest for the 17th is, for me, extremely relatable content. I don’t say that my interest in mathematics is entirely because there was this Berenstain Bears book about jobs which made it look like a mathematician’s job was to do sums in an observatory on the Moon. But it didn’t hurt. When I joke about how seven-year-old me wanted to be the astronaut who drew Popeye, understand, that’s not much comic exaggeration.

Student in mathematics class: 'I'd like a career where I solve simple subtraction problems like this. I'd be making a difference.'
Thaves’s Frank and Ernest for the 17th of October, 2018. “I’d be getting taped to the doors of mathematics teacher’s classrooms — saaaaaay!”

Justin Thompson’s Mythtickle rerun for the 17th is a timely choice about lotteries and probabilities. Vlad raises a fair point about your chance of being struck by lightning. It seems like that’s got to depend on things like where you are. But it does seem like we know what we mean when we say “the chance you’ll be hit by lightning”. At least I think it means “the probability that a person will be hit by lightning at some point in their life, if we have no information about any environmental facts that might influence this”. So it would be something like the number of people struck by lightning over the course of a year divided by the number of people in the world that year. You might have a different idea of what “the chance you’ll be hit by lightning” means, and it’s worth trying to think what precisely that does mean to you.

Dziva: 'Lottery tickets? Are you bats? Vlad, your chances of getting hit by lightning are better than winning a lottery jackpot.' Vlad: 'Lightning where? The location of the lightning is a variable that should be included in your determination. So do you mean like, lightning in the Atacama Desert where it never rains, or like lightning in, say, Transylvania? Cause back home, let me tell ya ... ' Dziva: 'Oh, I got it, I got it, I ... um.' [ Transylvanian convenience store with a werewolf cashier; lightning outside. ] Dziva: 'Two computer lotto-picks on the BIG one and make it SNAPPY, wolfie!!!'
Justin Thompson’s Mythtickle rerun for the 17th of October, 2018. Not to step on a joke Thompson left nicely underplayed, but I find funny the premise that of course the clerk in the Transylvanian convenience store is a werewolf. I have no information about when this strip first ran.

Lotteries are one of those subjects that a particular kind of nerd likes to feel all smug about. Pretty sure every lottery comic ever has drawn a comment about a tax on people who can’t do mathematics. This one did too. But then try doing the mathematics. The Mega Millions lottery, in the US, has a jackpot for the first drawing this week estimated at more than a billion dollars. The chance of winning is about one in 300 million. A ticket costs two dollars. So what is the expectation value of playing? You lose two dollars right up front, in the cost of the ticket. What do you get back? A one-in-300-million chance of winning a billion dollars. That is, you can expect to get back a bit more than three dollars. The implication is: you make a profit of dollar on each ticket you buy. There’s something a bit awry here, as you can tell from my decision not to put my entire savings into lottery tickets this week. But I won’t say someone is foolish or wrong if they buy a couple.

Student, to the teacher in front of a blackboard full of symbols: 'Can't you just round it off?'
Mike Baldwin’s Cornered for the 18th of October, 2018. What is that grit on the teacher’s desk, to the reader’s right of the pen?

Mike Baldwin’s Cornered for the 18th is a bit of mathematics-circling wordplay, featuring the blackboard full of equations. The blackboard doesn’t have any real content on it, but it is a good visual shorthand. And it does make me notice that rounding a quantity off is, in a way, making it simpler. If we are only a little interested in the count of the thing, “two thousand forty” or even “two thousand” may be more useful than the exact 2,038. The loss of precision may be worth it for the ease with which the rounded-off version is remembered and communicated.


If you’d like to see more Reading the Comics posts then try this link. Other essays which mention Archie should be at this link. Topics raised by Wallace the Brave should be at this link. Frank and Ernest is the subject of essays at this link. Topics brought up by Mythtickle are at this link. It’s a new tag, though, and I’m not sure there’ll ever be another use of it. And this and other essays mentioning Cornered are at this link. And do please stick around for more of my Fall 2018 Mathematics A-To-Z, coming twice a week through the rest of the year, I hope.

Reading the Comics, October 14, 2018: Possessive Edition


The first two comics for this essay have titles of the form Name’s Thing, so, that’s why this edition title. That’s good enough, isn’t it? And besides this series there was a Perry Bible Fellowship which at least depicted mathematical symbols. It’s a rerun, though, even among those shown on GoComics.com. It was rerun recently enough that I featured it around here back in June. It’s a bit risque. But the strip was rerun the 12th. Maybe I also need to drop Perry Bible Fellowship from the roster of comics I read for this.

On to the comics I haven’t dropped.

Tony Buino and Gary Markstein’s Daddy’s Home for the 11th tries using specific examples to teach mathematics. There’s strangeness to arithmetic. It’s about these abstract things like “thirty” and “addition” and such. But these things match very well the behaviors of discrete objects, ones that don’t blend together or shatter by themselves. So we can use the intuition we have for specific things to get comfortable working with the abstract. This doesn’t stop, either. Mathematicians like to work on general, abstract questions; they let us answer big swaths of questions all at once. But working out a specific case is usually easier, both to prove and to understand. I don’t know what’s the most advanced mathematics that could be usefully practiced by thinking about cupcakes. Probably something in group theory, in studying the rotations of objects that are perfectly, or nearly, rotationally symmetric.

Dad: 'It's like this: if mom made 30 cupcakes, and you gave 12 friends two cupcakes each, how many would you have left?' Elliot: 'Cupcakes or friends?' Dad: 'Good question.'
Tony Buino and Gary Markstein’s Daddy’s Home for the 11th of October, 2018. The coloring makes the strip more visually interesting than just the sketchy line art would; I’m curious how it’s rendered in newspapers that print black-and-white comics.

John Zakour and Scott Roberts’s Maria’s Day for the 11th is a follow-up to a strip featured last week. Maria’s been getting help on her mathematics from one of her closet monsters. And includes the usual joke about Common Core being such a horrible thing that it must come from monsters. I don’t know whether in the comic strip’s universe the monster is supposed to be imaginary. (Usually, in a comic strip, the question of whether a character is imaginary-or-real is pointless. I think Richard Thompson’s Cul de Sac is the only one to have done something good with it.) But if the closet monster is in Maria’s imagination, it’s quite in line for her to think that teaching comes from some malevolent and inscrutable force.

Maria: 'OK, you helped me with my homework, and I brought you a whole package of hot dogs. You wouldn't gobble me up after all that, right?' Monster: 'Kid, those were tofu dogs. I promise nothing. 'Course, I do wanna hear how I did on the math. Y'know, monsters invented 'common core'.'
John Zakour and Scott Roberts’s Maria’s Day for the 11th of October, 2018. Yeah, snarking on Common Core is an easy joke, but then so is snarking on tofu dogs. I’ve been happy with Morningstar vegetarian hot dogs for years now, although I will admit we don’t usually have them as hot dogs but instead as ways to bulk up a macaroni and cheese or similar meal.

Olivia Jaimes’s Nancy for the 12th features one of the first interesting mathematics questions you do in physics. This is often done with calculus. Not much, but more than Nancy and Esther could realistically have. It could be worked out experimentally, and that’s likely what the teacher was hoping for. Calculus isn’t really necessary, although it does show skeptical students there’s some value in all this d-dx business they’ve been working through. You can find the same answers by dimensional analysis, which is less intimidating. But you’d still need to know some trigonometry functions. That’s beyond whatever Nancy’s grade level is too. In any case, Nancy is an expert at identifying unstated assumptions, and working out loopholes in them. I’m curious whether the teacher would respect Nancy’s skill here. (The way the writing’s been going, I think she would.)

Teacher: 'Your assignment is to figure out what release angle makes a thrown ball travel the farthest.' Nancy, at the top of a well: 'Straight down seems to work pretty well.'
Olivia Jaimes’s Nancy for the 12th of October, 2018. Nancy and Esther found similarly good results at Bottomless Chasm, on the edge of town.

Francesco Marciuliano and Jim Keefe’s Sally Forth for the 13th is about new-friend Jenny trying to work out her relationship with Hilary-Faye-and-Nona. It’s a good bit of character work, but that is outside my subject here. In the last panel Nona admits she’s been talking, or at least thinking about τ versus π. This references a minor nerd-squabble that’s been going on a couple years. π is an incredibly well-known, useful number. It’s the only transcendental number you can expect a normal person to have ever heard of. Humans noticed it, historically, because the length of the circumference of a circle is π times the length of its diameter. Going between “the distance across” and “the distance around” turns out to be useful.

Jenny: 'You know, actions speak louder than words. And you three talk a *lot*. But ... I'll see where this goes, and if you mean what you say ... so, what were you talking about?' Hilary: 'Oddly enough, it never seems to be about school.' Nona: 'I was thinking about tau versus pi. But that might have just been in my head.'
Francesco Marciuliano and Jim Keefe’s Sally Forth for the 13th of October, 2018. I like the composition in the first panel. It’s the rare cinematic angle that allows all four people who need to be in the panel to be shown without looking like a police lineup. (Which, yeah, the third panel kind of does. But I don’t know how to frame that so you can show all four characters and have the three talking, and have the dialogue balloons read in the logical order.)

The thing is, many mathematical and physics formulas find it more convenient to write things in terms of the radius of a circle or sphere. And this makes 2π show up in formulas. A lot. Even in things that don’t obviously have circles in them. For example, the Gaussian distribution, which describes how much a sample looks like the population it’s sampled from, has 2π in it. So, the τ argument does, why write out 2π all these places? Why not decide that that’s the useful number to think about, give it the catchy name τ, and use that instead? All the interesting questions about π have exact, obvious parallel questions about τ. Any answers about one give us answers about the other. So why not make this switch and then … pocket the savings in having shorter formulas?

You may sense in me a certain skepticism. I don’t see where changing over gets us anything worth the bother. But there are fashions in mathematics as with everything else. Perhaps τ has some ability to clarify things in ways we’ll come to better appreciate.


This and my other Reading the Comics posts are this link. Essays inspired by Daddy’s Home are at this link. Other essays that mention Maria’s Day discussions should be at this link. Essays with a mention of Nancy, old and new, are at this link. And essays in which Sally Forth gets discussed will be at this link. It’s a new tag today, which does surprise me.

Reading the Comics, October 11, 2018: Under Weather Edition


I ended up not finding more comics on-topic on GoComics yesterday. So this past week’s mathematically-themed strips should fit into two posts well. I apologize for any loss of coherence in this essay, as I’m getting a bit of a cold. I’m looking forward to what this cold does for the A To Z essays coming Tuesday and Friday this week, too.

Stephen Beals’s Adult Children for the 7th uses Albert Einstein’s famous equation as shorthand for knowledge. I’m a little surprised it’s written out in words, rather than symbols. This might reflect that E = mc^2 is often understood just as this important series of sounds, rather than as an equation relating things to one another. Or it might just reflect the needs of the page composition. It could be too small a word balloon otherwise.

(In a darkened bar.) Harvey: 'Are they going to close?' Berle: 'They'll have to if this isn't fixed.' Harvey: 'E equals MC squared!' (The lights come on.) Berle 'Yay! ... Why did you yell that?' Harvey: 'Knowledge is power.'
Stephen Beals’s Adult Children for the 7th of October, 2018. I’m still not sure how I feel about this strip’s use of slightly offset panels like this.

Julie Larson’s The Dinette Set for the 9th continues the thread of tip-calculation jokes around here. I have no explanation for this phenomenon. In this case, Burl is doing the calculation correctly. If the tip is supposed to be 15% of the bill, and the bill is reduced 10%, then the tip would be reduced 10%. If you already have the tip calculated, it might be quicker to figure out a tenth of that rather than work out 15% of the original bill. And, yes, the characters are being rather unpleasantly penny-pinching. That was just the comic strip’s sense of humor.

Burl: 'So a 15% tip four two Monte Cristo platters would be $1.26' Dale: 'So ours would be the same as yours ... waidda minute! Today is 10% OFF for AARP members! So we times our total by 25% to figure the tip?' Burl: 'It's simple, Dale. Take 10% off the $1.26 tip, which is 12.6 cents, round that up to 13 cents, the minus that fro $1.26, and her tip is now $1.13.' Dale: 'Wow! I'm impressed! You did all that in your head! I'll bet I woulda given too much!'
Julie Larson’s The Dinette Set for the 9th of October, 2018. This is a rerun; it originally ran the 2nd of December, 2007.

Todd Clark’s Lola for the 9th take the form of your traditional grumbling about story problems. It also shows off the motif of updating of the words in a story problem to be awkwardly un-hip. The problem seems to be starting in a confounding direction anyway. The first sentence isn’t out and it’s introducing the rate at which Frank is shedding social-media friends over time and the rate at which a train is travelling, some distancer per time. Having one quantity with dimensions friends-per-time and another with dimensions distance-per-time is begging for confusion. Or for some weird gibberish thing, like, determining something to be (say) ninety mile-friends. There’s trouble ahead.

Lola: 'Sammy boy. How's middle school going?' Sammy: 'Stupid story problems.' Lola: 'Whatcha got? Maybe I can help.' Sammy: 'If Frank unfriends people at a rate of six per hour while on a train travelling ... '
Todd Clark’s Lola for the 9th of October, 2018. Don’t let your eye be distracted by the coloring job done on Lola’s eyeglasses in the last panel there.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 10th proposes naming a particular kind of series. A series is the sum of a sequence of numbers. It doesn’t have to be a sequence with infinitely many numbers in it, but it usually is, if it’s to be an interesting series. Properly, a series gets defined by something like the symbols in the upper caption of the panel:

\sum_{i = 1}^{\infty} a_i

Here the ‘i’ is a “dummy variable”, of no particular interest and not even detectable once the calculation is done. It’s not that thing with the square roots of -1 in thise case. ‘i’ is specifically known as the ‘index’, since it indexes the terms in the sequence. Despite the logic of i-index, I prefer to use ‘j’, ‘k’, or ‘n’. This avoids confusion with that square-root-of-minus-1 meaning for i. The index starts at some value, the one to the right of the equals sign underneath the capital sigma; in this case, 1. The sequence evaluates whatever the formula described by a_i is, for each whole number between that lowest ‘i’, in this case 1, and whatever the value above the sigma is. For the infinite series, that’s infinitely large. That is, work out a_i for every counting number ‘i’. For the first sum in the caption, that highest number is 4, and you only need to evaluate four terms and add them together. There’s no rule given for a_i in the caption; that just means that, in this case, we don’t yet have reason to care what the formula is.

On the blackboard: 'Solve: 24 + 12 + 6 + 3 + ... = ?' He put down 48. Woman: 'I ... wow. You've never studied series and you got it instantly.' Man: 'The 'plus three dots' part means 'plus 3', right?' Caption: 'New sequence type: Lucky Moron sequences. Definition: any convergent series such that 3 + (the first four terms) = (the infinite series)'.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 10th of October, 2018. And if you think that’s a not-really-needed name for a kind of series, note that MathWorld has a definition for the “FoxTrot Series” based on one problem from the FoxTrot comic strip from 1998.

This is the way to define a series if we’re being careful, and doing mathematics properly. But there are shorthands, and we fall back on them all the time. On the blackboard is one of them: 24 + 12 + 6 + 3 + \cdots . The \cdots at the end of a summation like this means “carry on this pattern for infinitely many terms”. If it appears in the middle of a summation, like 2 + 4 + 6 + 8 + \cdots + 20 it means “carry on this pattern for the appropriate number of terms”. In that case, it would be 10 + 12 + 14 + 16 + 18 .

The flaw with this “carry on this pattern” is that, properly, there’s no such thing as “the” pattern. There are infinitely many ways to continue from whatever the start was, and they’re all equally valid. What lets this scheme work is cultural expectations. We expect the difference between one term and the next to follow some easy patterns. They increase or decrease by the same amount as we’ve seen before (an arithmetic progression, like 2 + 4 + 6 + 8, increasing by two each time). They increase or decrease by the same ratio as we’ve seen before (a geometric progression, like 24 + 12 + 6 + 3, cutting in half each time). Maybe the sign alternates, or changes by some straightforward rule. If it isn’t one of these, then we have to fall back on being explicit. In this case, it would be that a_i = 24 \cdot \left(\frac{1}{2}\right)^{i - 1} .

The capital-sigma as shorthand for “sum” traces to Leonhard Euler, because of course. I’m finding it hard, in my copy of Florian Cajori’s History of Mathematical Notations, to find just where the series notation as we use it got started. Also I’m not finding where ellipses got into mathematical notation either. It might reflect everybody realizing this was a pretty good way to represent “we’re not going to write out the whole thing here”.

With something marked 30% off. First customer: 'This is normally 100 bucks. How much will it be at 30% off?' Val: '$70.' First customer, to second: 'See, I told you percents are the same as dollars.' Second customer: 'When you're right, you're right.' Val, thinking: 'I pity the next retailer who has to convince him otherwise.'
Norm Feuti’s Retail for the 11th of October, 2018. Yes, the comments include people explaining how people doing Common Core mathematics would never find an answer to “30% off $20”. Also a commenter who explains how one would, probably do it: “30% of 10 is 3. 20 is twice 20, so, 30% off 20 would be twice 3, or 6. So, 20 minus 6, or 14.” Followed by someone saying that if you did it by real math, it would be .7 times 20.

Norm Feuti’s Retail for the 11th riffs on how many people, fundamentally, don’t know what percentages are. I think it reflects thinking of a percentage as some kind of unit. We get used to measurements of things, like, pounds or seconds or dollars or degrees or such that are fixed in value. But a percentage is relative. It’s a fraction of some original quantity. A difference of (say) two pounds in weight is the same amount of weight whatever the original was; why wouldn’t two percent of the weight behave similarly? … Gads, yes, I feel for the next retailer who gets these customers.

I think I’ve already used the story from when I worked in the bookstore about the customer concerned whether the ten-percent-off sticker applied before or after sales tax was calculated. So I’ll only share if people ask to hear it. (They won’t ask.)


When I’m not getting a bit ill, I put my Reading the Comics posts at this link. Essays which mention Adult Children are at this link. Essays with The Dinette Set discussions should be at this link. The essays inspired by Lola are at this link. There’s some mention of Saturday Morning Breakfast Cereal in essays at link, or pretty much every Reading the Comics post. And Retail gets discussed at this link.

Reading the Comics, October 6, 2018: Square Root of 144 Edition


And I have three last strips from last week to talk about. For those curious, I have ten comics for this week that I flagged for mention, at least before reading the Saturday GoComics pages. So that will probably be two or three installments next week. It’ll depend how many Saturday GoComics strips raise a point I feel like discussing.

Jim Toomey’s Sherman’s Lagoon for the 5th uses arithmetic as the archetypical homework problem that’s short enough to fit in a panel but also too hard for an adult to do. And, neatly, easy for a computer to do. Were I either shark here I’d have reasoned out the square root of 144 something like this: they’re not getting homework where they’d be asked the square root of something that wasn’t a perfect square. So it’s got to be a whole number. 144 is between 100 and 400, so it’s got to be the square root of something between 10 and 20. 144 is pretty close to 100, so 144’s square root is probably close to 10. The square of 1 is 1, so 11 squared has to be one-hundred-something-and-one. The square of 2 is 4, so 12 squared has to be one-hundred-something-and-four. The square of 3 is 9, so 13 squared has to be one-hundred-something-and-nine. The square of 4 is 16, so 14 squared has to be at least one-hundred-something-and-six. And by then we’re getting pretty far from 10. So the only plausible candidate is 12. Test that out and, what do you know, there it is.

Herman: 'Dad, can you help me with my math homework?' Sherman: 'Highly doubtful. But I'm sure Alexa can. Ask her anything.' Herman: 'Alexa, what's the square root of 144?' Alexa: 'The square root of 144 is 12.' Herman: 'Wow. She's good. has Mom finally decided to replace you?' Sherman: 'Should I be worried?'
Jim Toomey’s Sherman’s Lagoon for the 5th of October, 2018. And if you’re wondering how an Alexa eavesdropping device is working underwater, go away. The strip’s not for you, and that’s your loss, because it’s nicely low-key weird and teaches me more about ocean biology than I ever imagined I’d know.

Greg Cravens’s The Buckets for the 6th is a riff on the monkeys-at-keyboards joke. Well, what keeps monkeys-at-typewriters from writing interesting things is that they don’t have any selection. They just produce text to no end, in principle. Picking out characters and words that carry narrative is what makes essayists and playwrights. … That said, I think every instructor has faced the essay that is, somehow, worse than gibberish. The process is to try to find anything that could be credited, even if it’s just including at least one of the words from the topic of the essay, and move briskly on.

Eddie: 'You know the old saying about putting an infinite number of monkeys at an infinite number of typewriters, and eventually they'll accidentally write Shakespeare's plays?' Toby: 'I guess.' Eddie: 'My English teacher says that nothing about our class should worry those monkeys ONE BIT!'
Greg Cravens’s The Buckets for the 6th of October, 2018. I’m mostly sure the guy with all the hair is Eddie but, again, character lists. Please.

Larry Wright’s Motley for the 6th is a riff on the idea tips are impossibly complicated to calculate. And that any mathematics might as well be algebra. My question: what the heck calculation is Debbie describing here? There are different ways to find a 15 percent tip. One two-step one is to divide the bill by ten, which is easy and gets you 10 percent. Then divide that by two, which is not-hard, and gets you 5 percent. Add together the 10 percent and 5 percent and you get 15 percent. A one-step method is to just divide by six. This gets you a bit under 17 percent, but that’s close enough. It just requires an ability to divide by six.

Debbie, on the phone: '... multiply by two and move the decimal over one digit ... then divide by four and subtract that answer from the first answer.' Toady: 'Were you helping someone with their algebra?' Debbie: 'No, Dad's at the restaurant and wanted to leave a 15 percent tip.'
Larry Wright’s Motley for the 6th of October, 2018. This strip originally ran in 1987, although I can’t make out when. Also see earlier comments about cast lists; I had to look through about a month’s worth of comics to find both characters’ names here.

There’s other ways to go about it, surely. There are many ways to do any calculation you like. Some of them have the advantages of requiring fewer steps. Some require more steps, but hopefully easier steps. Debbie is, obviously, just describing a nonsensically complicated calculation, to fit the needs of the joke. I’m just trying to think of what a plausible process would lead into the first panel and still get the right answer.

My many Reading the Comics posts are at this link. Essays which mention Sherman’s Lagoon should be at this link. Other essays with The Buckets should appear at this link. And other essays discussing Motley Classics should be here.

Reading the Comics, October 6, 2018: Curve Edition


There’s three more comics from last week I want to talk about. To ease my workload I’m going to put those off until Saturday. This is not an attempt to inflate the number of posts I make so that I can do a post-a-day-for-a-month again, as has happened in previous A-to-Z series. I already missed yesterday anyway. I just didn’t have time to think of things to write about six comics yesterday.

Morrie Turner’s Wee Pals for the 3rd has an interesting description of a circle. Definitions are a big part of mathematical work. This is especially so as we tend to think of mathematical objects as things that relate to one another in different ways. You want a definition that includes the relationships that are important, and excludes the ones you don’t want.

Wellington: 'Can you describe a circle, Nipper?' Nipper: 'Sure! A circle is a curved line with no kinks in it ... its ends join up so as not to show where it began.'
Morrie Turner’s Wee Pals for the 3rd of October, 2018. It previously ran the 9th of October, 2013. I suspect the strip was a rerun even in 2013, as the lines are pretty tightly drawn; other strips around that era were much more erratic. Also in 2013 I don’t seem to have thought this strip worth mention, which shows how standards are mutating around these parts.

Nipper’s definition of a circle … well, eh. I wouldn’t say that captures a circle. A ‘closed smooth curve’, yes. It’s closed because the ends join up. It’s smooth because there aren’t any corners, any kinks in it. It’s a curve because … well, there you go. There are many interesting shapes that are closed smooth curves. You can find some by tossing a rubber band in the air and seeing what it looks like when it lands. But I think what most people find important about circles are ideas like all the points on a curve being the same distance from some single “center” point. Nipper would probably realize his definition didn’t work by experimenting. Try drawing shapes that meet the rule he set out, but that aren’t what he thinks a circle ought to be.

This can be fruitful. It can develop a sharper idea of what a definition ought to have. Or it might force you to accept, in order to get the cases you want included, that something which seems wrong has to count too. This mathematicians faced in the late 19th and early 20th centuries. We learned that the best definition we’ve had for an idea like “a continuous function” means we have to allow weird conclusions, like that it’s possible to have a function continuous at a single point and nowhere else. But any other definition rules out things we absolutely have to call continuous, so, what’s there to do?

(Flo and a friend watch Flo's teenage daughter and friend walk past.) Friend: 'Sometimes I look at those girls and wish I was that age again.' Flo: 'I can understand that. But then you remember how rich your life has been?' Friend: 'No. Then I remember the algebra.'
Jenny Campbell’s Flo and Friends for the 4th of October, 2018. I’m not sure which friend Flo is speaking with here; Wikipedia suggests either Ruthie or Winnie most likely. So if I may summarize: cast lists. Why do we not have them? Thank you.

Jenny Campbell’s Flo and Friends for the 4th presents algebra as one of the burdens of youth. And one that’s so harsh that it makes old age more pleasant. I get the unpleasantness of being stuck in a class one doesn’t understand or like. But my own slight experience with that thing where you wake up, and a thing hurts, and there’s no good reason but eventually it either goes away or you get so used to it you don’t realize it still actually hurts? I would take the boring class, most of the time.

Maria's Mother: 'Very good, Maria! You got every problem on your math homework right.' (Later.) Maria: 'Thanks for the help, Math Wiz!' Math-wiz monster in the closet: 'Yeah, yeah. Where's my raw chicken? A deal's a deal.'
John Zakour and Scott Roberts’s Maria’s Day for the 4th of October, 2018. All right, so she got the questions right, but we can see she didn’t write her name at the top of the page. Please, please, PLEASE make sure you put your name on the page. Every page, too. It’s so very stressful for your instructor to have to figure out which of the three anonymous papers are which.

John Zakour and Scott Roberts’s Maria’s Day for the 4th is a joke about how hard mathematics is. Maria’s finding the monsters in her room less frightening than arithmetic. Well, as long as she’s picking up a couple useful things about multiplication.


I do at least one Reading the Comics post per week, and often two.They’ll be at this link. Other appearances by Wee Pals should be at this link. Topics raised by Flo and Friends are discussed at this link. And essays mentioning Maria’s Day are at this link. Thanks as ever for reading. I’m trusting that you did, or you wouldn’t be seeing this.

Reading the Comics, October 2, 2018: Frazz Loves Mathematics Edition


Jef Mallet’s Frazz did its best to take over my entire Reading-the-Comics bit this week. I won’t disrespect his efforts, especially as I take the viewpoint of the strip to be that arithmetic is a good thing to learn. Meanwhile let me offer another mention of Playful Mathematics Education Blog Carnival #121, hosted here last week. And to point out the Fall 2018 Mathematics A To Z continues this week with the letters ‘E’ and ‘F’. And I’m still looking for topics to discuss for select letters between H and M yet.

Sandra Bell-Lundy’s Between Friends for the 1st is a Venn Diagram joke to start off the week. The form looks wrong, though. This can fool the reader into thinking the cartoonist messed up the illustration. Here’s why. The point of a Venn Diagram is to show the two or more groups of things and identify what they have in common. It is true that any life will have regrets about things done. And regrets about things not done. But what are the things that one both ‘did do’ and ‘didn’t do’? Unless you accept the weasel-wording of “did halfheartedly”, there is nothing that one both did and did not.

[ The Venn Diagram of Living your Best Life ] The balloons are 'Did Do' and 'Didn't Do', with the overlap 'Regret'.
Sandra Bell-Lundy’s Between Friends for the 1st of October, 2018. Regardless of the internal logic of the joke, I like the art touch of the protagonist leaning against one of the bubbles. It makes the joke more interesting than simply being a pair of bubbles would be.

And here is where I will argue Bell-Lundy did this right. The overlap of things one ‘did do’ and ‘didn’t do’ must be empty. Do not be fooled by there being area in common in the overlap. One thing Venn Diagrams help us establish are the different kinds of things we are studying, and to work out whether that kind of thing can have any examples. And if the set of things in your life that you regret is empty — well! Is it not “living your best life”, as the caption advances, to have nothing one regrets doing, and nothing one regrets not doing? Thus I say to you the jury of readers, Sandra Bell-Lundy has correctly used the Venn Diagram form to make a “No Regrets” art.

That said, I can’t explain why the protagonist on the left is slumping and looking depressed. I suppose we have to take that she hasn’t lived her best life, but does have information about what might have been.

Jeff Mallet’s Frazz for the 1st starts a string of mathematics class jokes. Here is one about story problems, particularly ones about pricing apples and groups of apples. I don’t know whether apples are used as story problem examples. They seem like good example objects. They’re reasonably familiar. A person can have up to several dozen of them without it being ridiculously many. (Count a half-bushel of apples sometime.) You can imagine dividing them among people or tasks. You can even imagine halving and quartering them without getting ridiculous. Great set of traits. But the kid has overlooked that if Mrs Olsen wanted the price of an apple she would just look at the price sign.

Frazz: 'Very cool of you to bring treats.' Kid with box of doughnuts: 'Thanks; though it's mostly a story-problem prophylactic. If Mrs Olsen sees I went to the cider mill, she knows I know how much how many apples cost.' Frazz: 'Prophylactic!' Kid: 'If I can intimidate her out of a vocabulary unit, too, so much the better.'
Jeff Mallet’s Frazz for the 1st of October, 2018. I’m sorry, I don’t know the child’s name here.

(Every time I’m at the market I mean to check the apple prices, and I do, and I forget the total on the way out. I mention because I live in the same area as Jef Mallet. So there is a small but not-ridiculous chance he and I have bought apples from the same place. If he has a strip mentioning the place with the free coffee, popcorn, and gelato samples I’ll know to my satisfaction.)

Jeff Mallet’s Frazz for the 2nd has a complaint about having to show one’s work. But as with apple prices, we don’t really care whether someone has the right answer. We care whether they have the right method for finding an answer. Or, better, whether they have a method that could plausibly find the right answer, and an idea of how to check whether they did get it. This is why it’s worth, for example, working out a rough expected answer before doing a final calculation.

Frazz: 'How was math?' Kid: 'Same old argument: show your work. Do airline passengers care about the flight path? No! They care about the landing. Am I right?' Frazz: 'As right as anyone who's never had to scan for the barf bag while enjoying avoidable turbulence.'
Jeff Mallet’s Frazz for the 2nd of October, 2018. Seriously, why doesn’t every comic have an up-to-date cast roster with pictures? It would make everybody’s life so much easier.

The talk about flight paths reminds me of a story passed around sci.space.history back in the day. The story is about development of the automatic landing computers used for the Apollo Missions. The guidance computers were programmed to get the lunar module from this starting point to a final point on the lunar surface. This turns into a question of polynomial interpolation. That’s coming up with a curve that fits some data points, particularly, the positions and velocities the last couple times those were known plus the intended landing position. You can always find a polynomial that passes smoothly through a finite bunch of data points. That’s not hard. But, allegedly, the guidance computer would project paths where the height above the lunar surface was negative for a while. Numerically, there’s nothing wrong with a negative number. It’s just got some practical problems, as the earliest Apollo missions were before any subway tunnels could be built.

Kid: 'Why do I have to show my work in math when I know the right answer? I don't have to show my work in, say, writing.' Frazz: 'Writing *is* showing your work.' Kid: 'Spelling, then.' Frazz: 'Spelling is to writing what addition and multiplication tables are to algebra.'
Jeff Mallet’s Frazz for the 3rd of October, 2018. Also while writing does show off your work, it doesn’t show all your work, not if you’re writing anything that isn’t really basic. There’s research, there’s thinking, and there’s (often) outlines and drafts and revisions and editing. And, come to think of my own school experience, I didn’t like showing my outlines or drafts either. (In my defense, I am one of those people with a gift for academic narrative, so I can do an essay of up to two thousand words with little to no outline. And while I do better on my second draft, I don’t do enormously better most of the time.)

Jeff Mallet’s Frazz for the 3rd continues the protest against showing one’s work. I do like the analogy of arithmetic skills for mathematics being like spelling skills for writing. You can carry on without these skills, for either mathematics or writing. But knowing them makes your life easier. And enjoying these building-block units foreshadows enjoying the whole. But yeah, addition and multiplication tables can look like tedium if you don’t find something at least a little thrilling in how, say, 9 times 7 is 63.

Executive in a business office, speaking to an underling: 'We need to investigate an ADD cluster in accounting.'
Tim Lachowski’s Get a Life for the 2nd of October, 2018. You know, that guy has an enormous desk. Maybe he’s at a conference table.

Tim Lachowski’s Get a Life for the 2nd is a bit of mathematics wordplay. So that closes the essay out well.

Thanks for reading Reading the Comics. Other comic strip review essays are at this link. More essays with Between Friends should be at this link. Other essays with Frazz in them are at this link. And appearances by Get A Life should be at this link.

Reading the Comics, September 29, 2018: Vintage Comics Edition


Four more comics from last week struck me as worth mentioning. Two of them are over sixty years old.

Incidentally, Walt Kelly’s Pogo first appeared in the newspaper seventy years ago today. I don’t know anyone rerunning the comics the way Skippy or Thimble Theatre (Popeye) or the like are, which is a shame. (Few if any strips would be on-point around here, but it’s still worth reading.) But I did think some of the folks around here would like to know.

Percy Crosby’s Skippy for the 25th is a vintage-1931 strip about the miseries of learning arithmetic. Skippy’s scheme to both improve by copying one another’s 50-percent-right papers is not necessarily a bad one. It depends on a couple things to work. For example, do they both get the same questions wrong? Possibly; it’d be natural for both students to do worse on the harder questions. But suppose that the questions Skippy and Sooky get wrong are independent of one another. That is, knowing that Skippy got a question right doesn’t affect our estimate of the probability whether Sooky got that question right. In that case, we’d expect both of them to get about 25% of the questions right. And at least one of them would get about 75% of the questions right. So, if they could copy the right answers, they could get a 25-point improvement. That’s pretty good.

Skippy: 'Fifty in arithmetic again.' Sookie: 'That's funny, I got fifty, too.' Skippy: 'I got a scheme, you copy from me an' I'll copy from you. But we must be very careful to copy the right answers.'
Percy Crosby’s Skippy for the 25th of September, 2018. It originally ran, looks like, the 28th of May, 1931. Skippy’s the one talking; I’ve been calling the other kid Sooky, but am not confident I’m right the way I’m sure Charlie Brown is talking to Violet below.

Telling which are the right answers is hard. But, it’s typically easier to check whether an answer is right than it is to find an answer. Arithmetic is a point where this might not be usefully so. You can verify that 25 – 17 is indeed 8 by trying to calculate 17 + 8. But I don’t know that one equation is easier than the other.

Cynthia: 'Miss Lanham, I have a question about the assignment.' Lanham: 'Of course you do. Go ahead.' Cynthia: 'How much does it count toward our grade?' Lanham: 'About five percent.' Cynthia: 'Good. That's about the amount of effort I was figuring on expending.' Lanham: 'You only get the five percent with 100 percent effort.' Cynthia: 'That sounds like an energy hog.'
Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde for the 26th of September, 2018. Not depicted: how long it takes students to understand the course grade is a weighted average.

Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde for the 26th is a percentages joke. Miss Latham is making the supposition that one hundred percent effort is needed to get the assignment done correctly. That’s fair if the full effort to make is “what effort it takes to do the assignment correctly”. Tautological, but indisputable. If the one-hundred-percent-effort is whatever’s considered the appropriate standard effort to make for an assignment this size … well, that’s harder to agree with. Some assignments, some days, are easy; some just aren’t. Depends on what’s being asked.

Bill Schorr’s The Grizzwells for the 27th says it’s about mathematics. The particular question is about how many quarts go into a gallon. Measurement questions like this do get bundled into mathematics. It’s a bit hard to say why, though. It’s arbitrary how big a unit is; all we really demand is that it be convenient for whatever we’re doing. It’s even more arbitrary what the subdivisions of a unit are. A quart — well, the name gives away, it should be a quarter of something bigger. But there’s no reason we couldn’t have divided a gallon into three pieces, or six, or twelve instead. We just didn’t happen to do that. And similarly for subdividing a quart (or whatever name it would get, if it were a sixth of a gallon).

Pierpoint: 'Here's a math quiz. How many quarts are there in a gallon?' Gunther: 'I'd have to see the size of the gallon first.' Pierpoint: 'How did you know that?' Gunther: 'I've always been a bit of an algebra buff.'s
Bill Schorr’s The Grizzwells for the 27th of September, 2018. The bear is named Gunther and the porcupine Pierpoint. I mention this mostly for my own benefit. I’ve had to look up their names a surprising amount and it’s always harder than it should be. Every comic strip site needs a page that offers an up-to-date cast roster, including pictures, names, and relationships to the other main characters.

I suppose it’s from thinking of arithmetic as a tool for clerks and shopkeepers. These calculations would need to carry along units. Even the currency might need to carry units. Decimal currency obscures the units. Older-style pound-shilling-pence units (or whatever they were called in the local language) don’t allow that. So I’m guessing that it was natural to think of, say, “quadruple three quarts” as the same sort of problem as “one-sixth of 8s/4d”.

Charlie Brown, showing off a circle drawn on a fence: 'How's that? A PERFECT circle!' Violet: 'Uh huh ... what other kind of circles are there?' (Charlie Brown is silent.)
Charles Schulz’s Peanuts Begins for the 29th of September, 2018. It originally ran the 29th of June, 1954.. Violet was one of the original cast, but she did pretty much disappear from the strip when it turned out Lucy and Peppermint Patty were way better characters. She last appeared in a 1992 strip that got rerun during Schulz’s little hiatus in late 1997 so good luck giving an un-challengeable statement of her last appearance in the pre-eternal-reruns comic strip run.

Charles Schulz’s Peanuts Begins for the 29th speaks of “a perfect circle”. Violet asks an excellent question. But to say “a perfect circle” does communicate something. We name things like circles and lines and squares and agree they have certain properties. Also that the circles or lines or squares that we see in the world don’t have those properties. We might emphasize that something is a perfect circle or a straight line or something, to insist that it approaches this ideal of circle-ness. I’m not well-versed in the philosophy of mathematics. But it does seem hard to avoid Platonist thoughts about it. It’s hard to do geometry without pictures. But we insist to ourselves that the pictures may lie to us.

My other Reading the Comics posts should appear at this link. Percy Crosby’s Skippy gets mentions in essays at this link. There’s not many of them, but I really like the strip, so I hope there’s chances for more soon. Essays discussing topics raised by Barney and Clyde are at this link. Essays which discuss The Grizzwells are at this link. And Peanuts — both the 1970s “current” runs syndicated to newspapers and the 1950s “vintage” rerun only online — are at this link. And please stick around; there’ll be another A to Z post in about a day unless things go wrong.

Reading the Comics, September 24, 2018: Carnival Delay Edition


It’s unusual for me to have a Reading the Comics post on Monday, but that’s what fits my schedule. The Playful Mathematics Education Blog Carnival took my Sunday spot, and Tuesday and Friday I hope to continue the A to Z posts. It’s going to be a rather full week. I’m looking forward to, I hope, surviving. Meanwhile, here’s some comics.

Mike Thompson’s Grand Avenue for the 23rd resumes its efforts to become my archenemy with a strip about why learn arithmetic. Michael is right that we don’t need people to do multiplication. So why should we learn it? Grandmom Kate offers only the answer that he’ll be punished if he doesn’t learn them. This could motivate Michael to practice multiplication tables. But it’ll never convince him that learning multiplication tables is something of value.

Michael: 'You once told me: what you don't know won't hurt you. So I figure that means I don't need to know my multiplication tables!' Grandmom: 'You're right. Not knowing your multiplication tables won't hurt you. It will hurt both you and your allowance.' Michael: 'Wait! You're saying there's a possibility I might not get my allowance?' Grandmom: 'What you don't know won't hurt you.'
Mike Thompson’s Grand Avenue for the 23rd of September, 2018. So, not to nitpick the writing of these strips. Comic strip dialogue is subject to a great number of constraints that aren’t obvious, and that constrain how naturalistic it can be. But the dialogue flow is a slight wreck. Third panel to fourth, Grandmom goes from saying not knowing the multiplication tables won’t hurt you to saying it will hurt you and your allowance. Then, fourth panel to fifth, Michael goes from hearing he and his allowance will be hurt to wondering if there’s a possibility he might not get his allowance. It’s like they’re having different drafts of the conversation.

That said, what would convince him? It’s ridiculous to suppose Michael would be in a spot where he’d need to know eight times seven right away and without a computer to tell him. I find a certain amount of arithmetic-doing fun. But I already like doing it. (I admit a bootstrapping problem. Do I find it fun because I do it well, or do I do arithmetic well because I find it fun? I don’t know.) And that I find something fun is a lousy argument that everyone should learn to do it. I can argue that practicing multiplication tables is practice for finding neat patterns in other things, in higher mathematics. But is that reason to care? If Michael isn’t interested in eight times seven, is he going to be interested in the outer products of the set of symmetries on the octagon and the permutations of the heptagon?

I don’t have an actual answer here. I think it’s worth learning to do arithmetic. But not because we need people to do arithmetic. At least not except when we’re too lazy to take out our phones. But “or else you’ll lose money” is a terrible reason.

Things You Should Never, Ever Say To A Cartoonist. (It's nine panels of things they likely hear often, including 'Do you ever do any *real* art?' and 'You should try making your comics funnier' and 'You should do comics about my family, let me tell you about them!' Only one sends him running.) Questioner: 'What's the square root of 64 ? Why are you running away? Why are you screaming?' Corner Squirrel: 'Wait! Where do you get your ideaaaaas?!'
Dave Whamond’s Reality Check for the 23rd of September, 2018. That fourth panel happens to me, by the way, anytime I recount some odd, slightly awkward interaction that I have. And my family heritage is such that I’m a carrier of awkward interactions. Not to brag, but include a Nebus in your social circle and you’re more than ten times more likely than you ever imagined you could be to have responsibility for a styrofoam-peanut whirlwind, for example.

Dave Whamond’s Reality Check for the 23rd is a smorgasbord strip of things cartoonists get told too often. It comes in here because I like the strip, and because the punch line is built in the fear of arithmetic. It’s traditional to think that cartoonists, as artists, haven’t got an interest in mathematics or science. I can’t deny that the time it takes to learn how to draw, and the focus it takes to make a syndication-worthy comic strip, hurt someone’s ability to study much mathematics. And vice-versa. But people are a varied bunch. Bill Amend, of FoxTrot, and Bud Grace, of the discontinued The Piranha Club, were both physics majors. Darrin Bell, of Candorville and Rudy Park, writes well about mathematical (and scientific) topics. Crockett Johnson, of the renowned 1940s comic strip Barnaby and the Harold and the Purple Crayon books, was literate enough in mathematics to do over a hundred paintings based on geometry theorems. Part of why I note when the mathematics put into the background of a strip is that I do like pointing out there’s no reason artists and mathematicians or scientists need to be separate people.

Teacher: 'Question number three: you have twelve apples, but your friend Timmy only has two. How do you convince Timmy that this is a fair arrangement?'
Tony Carrillo’s F Minus for the 24th of September, 2018. Question four is about how many people named Timmy are in class.

Tony Carrillo’s F Minus for the 24th uses the form of the story problem. This one of the classic form of apples distributed amongst people. The problem presented makes its politics bare. But any narrative, however thin, carries along with it cultural values. That mathematicians may work out things whose truth is (we believe) independent of the posed problem doesn’t mean the posed problem is universal.

Tortoise: 'Y'know, I'm embarrassed to tell you how old I am.' Dog: 'Can you whisper it?' Tortoise: 'I'll write my age in the sand real temporary-like.' Dog: 'Don't spell it out, I can't read words!' Tortoise: 'Those are Roman numerals!'
Steve Boreman’s Little Dog Lost rerun for the 24th of September, 2018. This is a rerun from the 8th of September, 2009. Little Dog Lost has ended its run, and of gentle comic strips it’s one that I particularly miss.

Steve Boreman’s Little Dog Lost rerun for the 24th is the Roman Numerals joke for the week. There is a connotation of great age to anything written in Roman Numerals. Likely because we are centuries past the time they were used for anything but ornament. And even in ornament they seem to be declining in age. I do wonder if the puniness of, say, ‘MMI’ or ‘MMXX’ as a sequence of numerals, compared to (say) ‘MCMXLVII’ makes it look better to just write ‘2001’ or ‘2020’ instead.


The full set of Reading the Comics posts should be at this link. Essays that discuss Grand Avenue should be at this link. This and other appearances by Reality Check should be at this link. Appearances by F Minus are at this link. And other essays with Little Dog Lost should be at this link. Thanks for reading along.