Reading the Comics, February 26, 2018: Possible Reruns Edition


Comic Strip Master Command spent most of February making sure I could barely keep up. It didn’t slow down the final week of the month either. Some of the comics were those that I know are in eternal reruns. I don’t think I’m repeating things I’ve already discussed here, but it is so hard to be sure.

Bill Amend’s FoxTrot for the 24th of February has a mathematics problem with a joke answer. The approach to finding the area’s exactly right. It’s easy to find areas of simple shapes like rectangles and triangles and circles and half-circles. Cutting a complicated shape into known shapes, finding those areas, and adding them together works quite well, most of the time. And that’s intuitive enough. There are other approaches. If you can describe the outline of a shape well, you can use an integral along that outline to get the enclosed area. And that amazes me even now. One of the wonders of calculus is that you can swap information about a boundary for information about the interior, and vice-versa. It’s a bit much for even Jason Fox, though.

Jef Mallett’s Frazz for the 25th is a dispute between Mrs Olsen and Caulfield about whether it’s possible to give more than 100 percent. I come down, now as always, on the side that argues it depends what you figure 100 percent is of. If you mean “100% of the effort it’s humanly possible to expend” then yes, there’s no making more than 100% of an effort. But there is an amount of effort reasonable to expect for, say, an in-class quiz. It’s far below the effort one could possibly humanly give. And one could certainly give 105% of that effort, if desired. This happens in the real world, of course. Famously, in the right circles, the Space Shuttle Main Engines normally reached 104% of full throttle during liftoff. That’s because the original specifications for what full throttle would be turned out to be lower than was ultimately needed. And it was easier to plan around running the engines at greater-than-100%-throttle than it was to change all the earlier design documents.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 25th straddles the line between Pi Day jokes and architecture jokes. I think this is a rerun, but am not sure.

Matt Janz’s Out of the Gene Pool rerun for the 25th tosses off a mention of “New Math”. It’s referenced as a subject that’s both very powerful but also impossible for Pop, as an adult, to understand. It’s an interesting denotation. Usually “New Math”, if it’s mentioned at all, is held up as a pointlessly complicated way of doing simple problems. This is, yes, the niche that “Common Core” has taken. But Janz’s strip might be old enough to predate people blaming everything on Common Core. And it might be character, that the father is old enough to have heard of New Math but not anything in the nearly half-century since. It’s an unusual mention in that “New” Math is credited as being good for things. (I’m aware this strip’s a rerun. I had thought I’d mentioned it in an earlier Reading the Comics post, but can’t find it. I am surprised.)

Mark Anderson’s Andertoons for the 26th is a reassuring island of normal calm in these trying times. It’s a student-at-the-blackboard problem.

Morrie Turner’s Wee Pals rerun for the 26th just mentions arithmetic as the sort of homework someone would need help with. This is another one of those reruns I’d have thought has come up here before, but hasn’t.

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Were Story Problems Ever Any Good?


I have been reading Mapping In Michigan and the Great Lakes Region, edited by David I Macleod, because — look, I understand that I have a problem. I just live with it. The book is about exactly what you might imagine from the title. And it features lots of those charming old maps where, you know, there wasn’t so very much hard data available and everyone did the best with what they had. So you get these maps with spot-on perfect Lake Eries and the eastern shore of Lake Huron looking like you pulled it off of Open Street Maps. And then Michigan looks like a kid’s drawing of a Thanksgiving turkey. Also sometimes they drop a mountain range in the middle of the state because I guess it seemed a little empty without.

The first chapter, by Mary Sponberg Pedley, is a biography and work-history of Louis Charles Karpinski, 1878-1956. Karpinski did a lot to bring scholastic attention to maps of the Great Lakes area. He was a professor of mathematics for the University of Michigan. And he commented a good bit about the problems of teaching mathematics. Pedley quoted this bit that I thought was too good not to share. It’s from Arithmetic For The Farm. It’s about the failure of textbooks to provide examples that actually reflected anything anyone might want to know. I quote here Pedley’s endnote:

Karpinski disparaged the typical “story problems” found in contemporary textbooks, such as the following: “How many sacks, holding 2 bushels, 3 pecks and 2 quarts each can be filled from a bin containing 366 bushels, 3 pecks, 4 quarts of what?” Karpinski comments: “How carefully would you have to fill a sack to make it hold 3 pecks 2 quarts of anything? And who filled the bin so marvelously that the capacity is known with an accuracy of one-25th of 1% of the total?” He recommended an easier, more practical means of doing such problems, noting that a bushel is about 1 & 1/4 or 5/4 cubic feet. Therefore the number of bushels in the bin is the length times width times 4/5; the easiest way to get 4/5 of anything is to take away one-fifth of it.

This does read to me like Pedley jumped a track somewhere. It seems to go from the demolition of the plausibility of one problem’s setup to demolishing the plausibility of how to answer a problem. Still, the core complaint is with us yet. It’s hard to frame problems that might actually come up in ways that clearly test specific mathematical skills.


And on another note. This is the 1,000th mathematical piece that I’ve published since I started in September of 2011. If I’m not misunderstanding this authorship statistic on WordPress, which is never a safe bet. I’m surprised that it has taken as long as this to get to a thousand posts. Also I’m surprised that I should be surprised. I know roughly how many days there are in a year. And I know I need special circumstances to post something more often than every other day. Still, I’m glad to reach this milestone, and gratified that there’s anyone interested in what I have to say. In my next thousand posts I hope to say something.

Wronski’s Formula For Pi: My Boring Mistake


Previously:


So, I must confess failure. Not about deciphering Józef Maria Hoëne-Wronski’s attempted definition of π. He’d tried this crazy method throwing a lot of infinities and roots of infinities and imaginary numbers together. I believe I translated it into the language of modern mathematics fairly. And my failure is not that I found the formula actually described the number -½π.

Oh, I had an error in there, yes. And I’d found where it was. It was all the way back in the essay which first converted Wronski’s formula into something respectable. It was a small error, first appearing in the last formula of that essay and never corrected from there. This reinforces my suspicion that when normal people see formulas they mostly look at them to confirm there is a formula there. With luck they carry on and read the sentences around them.

My failure is I wanted to write a bit about boring mistakes. The kinds which you make all the time while doing mathematics work, but which you don’t worry about. Dropped signs. Constants which aren’t divided out, or which get multiplied in incorrectly. Stuff like this which you only detect because you know, deep down, that you should have gotten to an attractive simple formula and you haven’t. Mistakes which are tiresome to make, but never make you wonder if you’re in the wrong job.

The trouble is I can’t think of how to make an essay of that. We don’t tend to rate little mistakes like the wrong sign or the wrong multiple or a boring unnecessary added constant as important. This is because they’re not. The interesting stuff in a mathematical formula is usually the stuff representing variations. Change is interesting. The direction of the change? Eh, nice to know. A swapped plus or minus sign alters your understanding of the direction of the change, but that’s all. Multiplying or dividing by a constant wrongly changes your understanding of the size of the change. But that doesn’t alter what the change looks like. Just the scale of the change. Adding or subtracting the wrong constant alters what you think the change is varying from, but not what the shape of the change is. Once more, not a big deal.

But you also know that instinctively, or at least you get it from seeing how it’s worth one or two points on an exam to write -sin where you mean +sin. Or how if you ask the instructor in class about that 2 where a ½ should be, she’ll say, “Oh, yeah, you’re right” and do a hurried bit of erasing before going on.

Thus my failure: I don’t know what to say about boring mistakes that has any insight.


For the record here’s where I got things wrong. I was creating a function, named ‘f’ and using as a variable ‘x’, to represent Wronski’s formula. I’d gotten to this point:

f(x) = -4 \imath x 2^{\frac{1}{2}\cdot \frac{1}{x}} \left\{ e^{\imath \frac{\pi}{4}\cdot\frac{1}{x}} -  e^{- \imath \frac{\pi}{4}\cdot\frac{1}{x}} \right\}

And then I observed how the stuff in curly braces there is “one of those magic tricks that mathematicians know because they see it all the time”. And I wanted to call in this formula, correctly:

\sin\left(\phi\right) = \frac{e^{\imath \phi} - e^{-\imath \phi}}{2\imath }

So here’s where I went wrong. I took the 4\imath way off in the front of that first formula and combined it with the stuff in braces to make 2 times a sine of some stuff. I apologize for this. I must have been writing stuff out faster than I was thinking about it. If I had thought, I would have gone through this intermediate step:

f(x) = -4 \imath x 2^{\frac{1}{2}\cdot \frac{1}{x}} \left\{ e^{\imath \frac{\pi}{4}\cdot\frac{1}{x}} -  e^{- \imath \frac{\pi}{4}\cdot\frac{1}{x}} \right\} \cdot \frac{2\imath}{2\imath}

Because with that form in mind, it’s easy to take the stuff in curled braces and the 2\imath in the denominator. From that we get, correctly, \sin\left(\frac{\pi}{4}\cdot\frac{1}{x}\right) . And then the -4\imath on the far left of that expression and the 2\imath on the right multiply together to produce the number 8.

So the function ought to have been, all along:

f(x) = 8 x 2^{\frac{1}{2}\cdot \frac{1}{x}} \sin\left(\frac{\pi}{4}\cdot \frac{1}{x}\right)

Not very different, is it? Ah, but it makes a huge difference. Carry through with all the L’Hôpital’s Rule stuff described in previous essays. All the complicated formula work is the same. There’s a different number hanging off the front, waiting to multiply in. That’s all. And what you find, redoing all the work but using this corrected function, is that Wronski’s original mess —

\pi = \frac{4\infty}{\sqrt{-1}}\left\{ \left(1 + \sqrt{-1}\right)^{\frac{1}{\infty}} -  \left(1 - \sqrt{-1}\right)^{\frac{1}{\infty}} \right\}

— should indeed equal:

2\pi

All right, there’s an extra factor of 2 here. And I don’t think that is my mistake. Or if it is, other people come to the same mistake without my prompting.

Possibly the book I drew this from misquoted Wronski. It’s at least as good to have a formula for 2π as it is to have one for π. Or Wronski had a mistake in his original formula, and had a constant multiplied out front which he didn’t want. It happens to us all.


Fin.

Reading the Comics, February 24, 2018: My One Boring Linear Algebra Anecdote Edition


Wait for it.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 21st mentions mathematics — geometry, primarily — as something a substitute teacher has tried teaching with the use of a cucumber and condom. These aren’t terrible examples to use to make concrete the difference between volumes and surface areas. There are limitations, though. It’s possible to construct a shape that has a finite volume but an infinitely large surface area, albeit not using cucumbers.

There’s also a mention of the spring constant, and physics. This isn’t explicitly mathematical. But the description of movement on a spring are about the first interesting differential equation of mathematical physics. The solution is that of simple harmonic motion. I don’t think anyone taking the subject for the first time would guess at the answer. But it’s easy enough to verify it’s right. And this motion — sine waves — just turns up everywhere in mathematical physics.

Bud Blake’s Tiger rerun for the 23rd just mentions mathematics as a topic Hugo finds challenging, and what’s challenging about it. So a personal story: when I took Intro to Linear Algebra my freshman year one day I spaced on the fact we had an exam. So, I put the textbook on the shelf under my desk, and then forgot to take it when I left. The book disappeared, of course, and the professor never heard of it being turned in to lost-and-found or anything. Fortunately the homework was handwritten questions passed out on photocopies (ask your parents), so I could still do the assignments, but for all those, you know, definitions and examples I had to rely on my own notes. I don’t know why I couldn’t ask a classmate. Shyness, probably. Came through all right, though.

Hugo: 'These math problems we got for homework are gonna be hard to do.' Tiger: 'Because you don't understand them?' Hugo: 'Because I brought home my history book by mistake.'
Bud Blake’s Tiger rerun for the 23rd of February, 2018. Whose house are they at? I mean, did Tiger bring his dog to Hugo’s, or did Hugo bring his homework to Tiger’s house? I guess either’s not that odd, especially if they just got out of school, but then Hugo’s fussing with his homework when he’s right out of school and with Tiger?

Cathy Law’s Claw for the 23rd technically qualifies as an anthropomorphic-numerals joke, in this panel about the smothering of education by the infection of guns into American culture.

Jim Meddick’s Monty for the 23rd has wealthy child Wedgwick unsatisfied with a mere ball of snow. He instead has a snow Truncated Icosahedron (the hyphens in Jarvis’s word balloon may baffle the innocent reader). This is a real shape, one that’s been known for a very long time. It’s one of the Archimedean Solids, a set of 13 solids that have convex shapes (no holes or indents or anything) and have all vertices the same, the identical number of edges coming in to each point in the same relative directions. The truncated icosahedron you maybe also know as the soccer ball shape, at least for those old-style soccer balls made of patches that were hexagons and pentagons. An actual truncated icosahedron needs twelve pentagons, so the figure drawn in the third panel isn’t quite right. At least one pentagonal face would be visible. But that’s also tricky to draw. The aerodynamics of a truncated icosahedron are surely different from those of a sphere. But in snowball-fight conditions, probably not different enough to even notice.

Mark Litzler’s Joe Vanilla for the 24th uses a blackboard full of formulas to represent an overcomplicated answer. The formulas look, offhand, like gibberish to me. But I’ll admit uncertainty since the odd capitalization of “iG(p)” at the start makes me think of some deeper group theory or knot theory symbols. And to see an “m + p” and an “m – p” makes me think of quantum mechanics of atomic orbitals. (But then an “m – p2” is weird.) So if this were anything I’d say it was some quantum chemistry formula. But my gut says if Litzler did take the blackboard symbols from anything, it was without going back to references. (Which he has no need to do, I should point out; the joke wouldn’t be any stronger — or weaker — if the blackboard meant anything.)

Reading the Comics, February 20, 2018: Bob the Squirrel Edition


So one comic strip was technically on point all this week, without ever quite giving me a specific thing to talk about. And I came to conclude there was another comic strip I could drop from my consideration. Which all were they? Read on.

Frank Page’s Bob the Squirrel for the 18th of February isn’t really about the Rubik’s Cube. It’s just something to occupy Bob’s mind until a deeper mystery emerges. Rubik’s Cubes, meanwhile, are everyone’s favorite group theory pastime, although I’m not sure how many people have learned group theory starting from that point. Where flies come from in the middle of winter I don’t know. We’ve been dealing with box elder bugs ourselves. (We’ve been scooping them up and tossing them outside where they can hopefully find the trees they should be using instead.)

Bob the Squirrel went on, during the week, to start a sequence about Lauren needing a geometry tutor. The story hasn’t done much that geometry-specific — Saturday’s was the most approximately on point — but it’s a comic strip I like. Squirrel fans might agree. (The strip for the 22nd has most tickled me.)

Allison Barrows’s PreTeena rerun for the 19th has a student teacher starting off her experience with a story problem. Your classic time-estimation problem.

Jack Pullan’s Boomerangs rerun for the 20th is one that mentions entropy and that I’ve already talked about at least twice before. These were times in January 2017 and also in November 2013. Given that the strip’s no longer in production and that I’m clearly on at least my third go-round I suppose I’ll retire it from my daily read. I’m curious why, if it was about 14 months between the last appearance and this appearance of this strip, why I didn’t have it at all in 2015 or 2016. Maybe I missed it, or it came a week there was enough to write about that I didn’t need to include a marginal strip.

Christopher Grady’s Lunarbaboon for the 20th is intended to be a heartwarming little story of encouragement and warm feelings. (Most Lunarbaboon strips are intended to be a heartwarming little story of encouragement and warm feelings.) That it’s mathematics the kid struggles with is incidental to the story setup. But it does make it easy to picture a kid struggling and a couple kind words offering some motivation, or at least better feelings.

Richard Thompson’s Richard’s Poor Almanac for the 20th is a casual mention of sudoku and a publication error that would supposedly have made it impossible. If the numbers were transposed consistently — everything that ought to have been a ‘2’ printed as a ‘5’, and everything that ought to have been ‘5’ printed as ‘2’ — the problem would be exactly as solvable. This is why you can sometimes see sudoku-type puzzles that use symbols or letters or other characters. But if, say, the third and the second rows were transposed then there’s a chance the incorrect puzzle would be solvable. Transposing a bunch of squares, like, the top three rows with the bottom three rows, wouldn’t make the puzzle unsolvable. This serves as a reminder that if you make enough mistakes you can still turn out all right, a comforting message for our times. Also I know I’ve featured Richard’s Poor Almanac several times over, but I’m a Richard Thompson fan so I’m not dropping that from my feed.

Will Henry’s Wallace the Brave — to be newspaper-syndicated from the 26th of March, by the way, and I’m glad for that as Wallace and I share the same favorite pinball game — just mentions mathematics as a subject Wallace isn’t thinking enough about. I’m also fond of the Loch Ness Monster, so, all the better.

I’m not surprised that this seems to be the first time I’ve had Lunarbabboon tagged. I am surprised that Bob the Squirrel seems not to have been tagged here before. Maybe I didn’t give the tag suggested-completion enough time to figure out what to do with ‘bob the’. We’ve been having odd little net glitches that mostly pass quickly, but that kill any sort of client-side Javascript-based page rendering. You know, like every web page does anymore because somehow “the web server puts together a bunch of stuff and transmits that to the reader” is too inefficient a system.

How Often Should Records Break? A Puzzle For You


Lansing got some record-breaking rain this week. Tuesday we got over two inches of rain, doubling the hundred-plus-year-old previous record. I mention because it got me to wondering how often we should expect records to break. I mean if the thing being measured probably isn’t changing. So my inspiration is out, as there’s no serious question about the climate changing. Measures of sports performance are also no good.

But we can imagine there’s something with an underlying property that isn’t changing. So if you keep getting samples of some independent, normally-distributed property in, how often should you expect to go between record-setting values? New records should start pretty thick on the ground. The first value is necessarily both a new high and low. The second is either a high or a low. The third seems to have a good chance of being a new extreme. Fourth, too. But somewhere along the way extremes should get rarer. Even if the 10,000th sample recorded is a new record high or low, what are the odds the 10,001st is? The 10,010th?

Haven’t got an answer offhand, although it’s surely available. Just mulling over how to attack the problem before I do what I always do and write a Matlab program to do a bunch of simulations. Easier than thinking. But I’ll leave the problem out for someone needing the challenge.

Reading the Comics, February 17, 2018: Continuing Deluge Month


February’s been a flooding month. Literally (we’re about two blocks away from the Voluntary Evacuation Zone after the rains earlier this week) and figuratively, in Comic Strip Master Command’s suggestions about what I might write. I have started thinking about making a little list of the comics that just say mathematics in some capacity but don’t give me much to talk about. (For example, Bob the Squirrel having a sequence, as it does this week, with a geometry tutor.) But I also know, this is unusually busy this month. The problem will recede without my having to fix anything. One of life’s secrets is learning how to tell when a problem’s that kind.

Patrick Roberts’s Todd the Dinosaur for the 12th just shows off an arithmetic problem — fractions — as the thing that can be put on the board and left for students to do.

Todd: *Sniff sniff* 'Hey! What's that on the floor?' (He follows a trail of beef jerky, eating, until he's at the chalkboard.) Teacher: 'Well, hello, Todd! Say, while you're up there, why don't you do that fractions problem on the board?' Todd: 'Darn you, tasty Slim jims!'
Patrick Roberts’s Todd the Dinosaur for the 12th of February, 2018. I’ll risk infecting you with one of my problems: I look at this particular comic and wonder what happened right before the first panel to lead to this happening.

Ham’s Life on Earth for the 12th has a science-y type giving a formula as “something you should know”. The formula’s gibberish, so don’t worry about it. I got a vibe of it intending to be some formula from statistics, but there’s no good reason for that. I’ve had some statistical distribution problems on my mind lately.

Eric Teitelbaum and Bill Teitelbaum’s Bottomliners for the 12th maybe influenced my thinking. It has a person claiming to be a former statistician, and his estimate of how changing his job’s affected his happiness. Could really be any job that encourages people to measure and quantify things. But “statistician” is a job with strong connotations of being able to quantify happiness. To have that quantity feature a decimal point, too, makes him sound more mathematical and thus, more surely correct. I’d be surprised if “two and a half times” weren’t a more justifiable estimate, given the margin for error on happiness-measurement I have to imagine would be there. (This seems to be the first time I’ve featured Bottomliners at least since I started tagging the comic strips named. Neat.)

Ruben Bolling’s Super-Fun-Pak Comix for the 12th reprinted a panel called The Uncertainty Principal that baffled commenters there. It’s a pun on “Uncertainty Principle”, the surprising quantum mechanics result that there are some kinds of measurements that can’t be taken together with perfect precision. To know precisely where something is destroys one’s ability to measure its momentum. To know the angular momentum along one axis destroys one’s ability to measure it along another. This is a physics result (note that the panel’s signed “Heisenberg”, for the name famously attached to the Uncertainty Principle). But the effect has a mathematical side. The operations that describe finding these incompatible pairs of things are noncommutative; it depends what order you do them in.

We’re familiar enough with noncommutative operations in the real world: to cut a piece of paper and then fold it usually gives something different to folding a piece of paper and then cutting it. To pour batter in a bowl and then put it in the oven has a different outcome than putting batter in the oven and then trying to pour it into the bowl. Nice ordinary familiar mathematics that people learn, like addition and multiplication, do commute. These come with partners that don’t commute, subtraction and division. But I get the sense we don’t think of subtraction and division like that. It’s plain enough that ‘a’ divided by ‘b’ and ‘b’ divided by ‘a’ are such different things that we don’t consider what’s neat about that.

In the ordinary world the Uncertainty Principle’s almost impossible to detect; I’m not sure there’s any macroscopic phenomena that show it off. I mean, that atoms don’t collapse into electrically neutral points within nanoseconds, sure, but that isn’t as compelling as, like, something with a sodium lamp and a diffraction grating and an interference pattern on the wall. The limits of describing certain pairs of properties is about how precisely both quantities can be known, together. For everyday purposes there’s enough uncertainty about, say, the principal’s weight (and thus momentum) that uncertainty in his position won’t be noticeable. There’s reasons it took so long for anyone to suspect this thing existed.

Samson’s Dark Side of the Horse for the 13th uses a spot of arithmetic as the sort of problem coffee helps Horace solve. The answer’s 1.

Mike Baldwin’s Cornered for the 14th is a blackboard-full-of-symbols panel. Well, a whiteboard. It’s another in the line of mathematical proofs of love.

Dana Simpson’s Ozy and Millie rerun for the 14th has the title characters playing “logical fallacy tag”. Ozy is, as Millie says, making an induction argument. In a proper induction argument, you characterize something with some measure of size. Often this is literally a number. You then show that if it’s true that the thing is true for smaller problems than you’re interested in, then it has to also be true for the problem you are interested in. Add to that a proof that it’s true for some small enough problem and you’re done. In this case, Ozy’s specific fallacy is an appeal to probability: all but one of the people playing tag are not it, and therefore, any particular person playing the game isn’t it. That it’s fallacious really stands out when there’s only two people playing.

Ed: 'Only recently, scientists discovered pigeons understand space and time.' Pigeon: 'They never questioned us before. We're waiting for them to ask us about the Grand Unified Theory of Physics next.'
Alex Hallatt’s Arctic Circle for the 16th of February, 2018. As ever, I learn things from doing this! Specifically the names of the penguins which I’d somehow not thought about before. Ed’s the one with a pair of antenna-like feathers on his head. Oscar has the smooth head. Gordo has the set of bumps.

Alex Hallatt’s Arctic Circle for the 16th riffs on the mathematics abilities of birds. Pigeons, in this case. The strip starts from their abilities understanding space and time (which are amazing) and proposes pigeons have some insight into the Grand Unified Theory. Animals have got astounding mathematical abilities, should point out. Don’t underestimate them. (This also seems to be the first time I’ve tagged Arctic Circle which doesn’t seem like it could be right. But I didn’t remember naming the penguins before so maybe I haven’t? Huh. Mind, I only started tagging the comic strip titles a couple months ago.)

Tony Cochrane’s Agnes for the 17th has the title character try bluffing her way out of mathematics homework. Could there be a fundamental flaw in mathematics as we know it? Possibly. It’s hard to prove that any field complicated enough to be interesting is also self-consistent. And there’s a lot of mathematics out there. And mathematics subjects often develop with an explosion of new ideas and then a later generation that cleans them up and fills in logical gaps. Symplectic geometry is, if I’m following the news right, going into one of those cleaning-up phases now. Is it likely to be uncovered by a girl in elementary school? I’m skeptical, and also skeptical that she’d have a replacement system that would be any better. I admire Agnes’s ambition, though.

Mike Baldwin’s Cornered for the 17th plays on the reputation for quantum mechanics as a bunch of mathematically weird, counter-intuitive results. In fairness to the TV program, I’ve had series run longer than I originally planned too.

Some Mathematics Things I Read On Twitter


I had thought I’d culled some more pieces from my Twitter and other mathematics-writing-reading the last couple weeks and I’m not sure where it all went. I think I might be baffled by the repostings of things on Quanta Magazine (which has a lot of good mathematics articles, but not, like, a 3,000-word piece every day, and they showcase their archive just as anyone ought).

So, here, first.

It reviews Kim Plofker’s 2008 text Mathematics In India, a subject that I both know is important — I love to teach with historic context included — and something that I very much bluff my way through. I mean, I do research things I expect I’ll mention, but I don’t learn enough of the big picture and a determined questioner could prove how fragile my knowledge was. So Plofker’s book should go on my reading list at least.

These are lecture notes about analysis. In the 19th century mathematicians tried to tighten up exactly what we meant by things like “functions” and “limits” and “integrals” and “numbers” and all that. It was a lot of good solid argument, and a lot of surprising, intuition-defying results. This isn’t something that a lay reader’s likely to appreciate, and I’m sorry for that, but if you do know the difference between Riemann and Lebesgue integrals the notes are likely to help.

And this, Daniel Grieser and Svenja Maronna’s Hearing The Shape Of A Triangle, follows up on a classic mathematics paper, Mark Kac’s Can One Hear The Shape Of A Drum? This is part of a class of problems in which you try to reconstruct what kinds of things can produce a signal. It turns out to be impossible to perfectly say what shape and material of a drum produced a certain sound of a drum. But. A triangle — the instrument, that is, but also the shape — has a simpler structure. Could we go from the way a triangle sounds to knowing what it looks like?

And I mentioned this before but if you want to go reading every Calvin and Hobbes strip to pick out the ones that mention mathematics, you can be doing someone a favor too.

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


And it’s not always fair to say that the gods mock any plans made by humans, but Comic Strip Master Command has been doing its best to break me of reading and commenting on any comic strip with a mathematical theme. I grant that I could make things a little easier if I demanded more from a comic strip before including it here. But even if I think a theme is slight that doesn’t mean the reader does, and it’s easy to let the eye drop to the next paragraph if the reader does think it’s too slight. The anthology nature of these posts is part of what works for them. And then sometimes Comic Strip Master Command sends me a day like last Sunday when everybody was putting in some bit of mathematics. There’ll be another essay on the past week’s strips, never fear. But today’s is just for the single day.

Susan Camilleri Konar’s Six Chix for the 11th illustrates the Lemniscate Family. The lemniscate is a shape well known as the curve made by a bit of water inside a narrow tube by people who’ve confused it with a meniscus. An actual lemniscate is, as the chain of pointing fingers suggests, a figure-eight shape. You get — well, I got — introduced to them in prealgebra. They’re shapes really easy to describe in polar coordinates but a pain to describe in Cartesian coordinates. There are several different kinds of lemniscates, each satisfying slightly different conditions while looking roughly like a figure eight. If you’re open to the two lobes of the shape not being the same size there’s even a kind of famous-ish lemniscate called the analemma. This is the figure traced out by the sun if you look at its position from a set point on the surface of the Earth at the same clock time each day over the course of the year. That the sun moves north and south from the horizon is easy to spot. That it is sometimes east or west of some reference spot is a surprise. It shows the difference between the movement of the mean sun, the sun as we’d see it if the Earth had a perfectly circular orbit, and the messy actual thing. Dr Helmer Aslasken has a fine piece about this, and how it affects when the sun rises earliest and latest in the year.

At a restaurant: 'It was always a challenge serving the lemniscate family'. Nine people each pointing to neighbors and saying 'I'll have what s/he's having', in a sequence that would make a figure-eight as seen from above or below the tables.
Susan Camilleri Konar’s Six Chix for the 11th of February, 2018. It’s not really worse than some of the Carioid Institute dinners.

There’s also a thing called the “polynomial lemniscate”. This is a level curve of a polynomial. That is, what are all the possible values of the independent variable which cause the polynomial to evaluate to some particular number? This is going to be a polynomial in a complex-valued variable, in order to get one or more closed and (often) wriggly loops. A polynomial of a real-valued variable would typically give you a boring shape. There’s a bunch of these polynomial lemniscates that approximate the boundary of the Mandelbrot Set, that fractal that you know from your mathematics friend’s wall in 1992.

Mark Anderson’s Andertoons took care of being Mark Anderson’s Andertoons early in the week. It’s a bit of optimistic blackboard work.

Lincoln Pierce’s Big Nate features the formula for calculating the wind chill factor. Francis reads out what is legitimately the formula for estimating the wind chill temperature. I’m not going to get into whether the wind chill formula makes sense as a concept because I’m not crazy. The thinking behind it is that a windless temperature feels about the same as a different temperature with a particular wind. How one evaluates those equivalences offers a lot of room for debate. The formula as the National Weather Service, and Francis, offer looks frightening, but isn’t really hard. It’s not a polynomial, in terms of temperature and wind speed, but it’s close to that in form. The strip is rerun from the 15th of February, 2009, as Lincoln Pierce has had some not-publicly-revealed problem taking him away from the comic for about a month and a half now.

Jim Scancarelli’s Gasoline Alley included a couple of mathematics formulas, including the famous E = mc2 and the slightly less famous πr2, as part of Walt Wallet’s fantasy of advising scientists and inventors. (Scientists have already heard both.) There’s a curious stray bit in the corner, writing out 6.626 x 102 x 3 that I wonder about. 6.626 is the first couple digits of Planck’s Constant, as measured in Joule-seconds. (This is h, not h-bar, I say for the person about to complain.) It’d be reasonable for Scancarelli to have drawn that out of a physics book or reference page. But the exponent is all wrong, even if you suppose he mis-wrote 1023. It should be 6.626 x 10-34. So I don’t know whether Scancarelli got things very garbled, or if he just picked a nice sciencey-looking number and happened to hit on a significant one. (There’s enough significant science numbers that he’d have a fair chance of finding something.) The strip is a reprint from the 4th of February, 2007, as Jim Scancarelli has been absent for no publicly announced reason for four months now.

Greg Evans and Karen Evans’s Luann is not perfectly clear. But I think it’s presenting Gunther doing mathematics work to support his mother’s contention that he’s smart. There’s no working out what work he’s doing. But then we might ask how smart his mother is to have made that much food for just the two of them. Also that I think he’s eating a potato by hand? … Well, there are a lot of kinds of food that are hard to draw.

Greg Evans’s Luann Againn reprints the strip from the 11th of February (again), 1990. It mentions as one of those fascinating things of arithmetic an easy test to see if a number’s a multiple of nine. There are several tricks like this, although the only ones anybody can remember are finding multiples of 3 and finding multiples of 9. Well, they know the rules for something being a multiple of 2, 5, or 10, but those hardly look like rules, and there’s no addition needed. Similarly with multiples of 4.

Modular arithmetic underlies all these rules. Once you know the trick you can use it to work out your own add-up-the-numbers rules to find what numbers are multiples of small numbers. Here’s an example. Think of a three-digit number. Suppose its first digit is ‘a’, its second digit ‘b’, and its third digit ‘c’. So we’d write the number as ‘abc’, or, 100a + 10b + 1c. What’s this number equal to, modulo 9? Well, 100a modulo 9 has to be equal to whatever a modulo 9 is: (100 a) modulo 9 is (100) modulo 9 — that is, 1 — times (a) modulo 9. 10b modulo 9 is (10) modulo 9 — again, 1 — times (b) modulo 9. 1c modulo 9 is … well, (c) modulo 9. Add that all together and you have a + b + c modulo 9. If a + b + c is some multiple of 9, so must be 100a + 10b + 1c.

The rules about whether something’s divisible by 2 or 5 or 10 are easy to work with since 10 is a multiple of 2, and of 5, and for that matter of 10, so that 100a + 10b + 1c modulo 10 is just c modulo 10. You might want to let this settle. Then, if you like, practice by working out what an add-the-digits rule for multiples of 11 would be. (This is made a lot easier if you remember that 10 is equal to 11 – 1.) And if you want to show off some serious arithmetic skills, try working out an add-the-digits rule for finding whether something’s a multiple of 7. Then you’ll know why nobody has ever used that for any real work.

J C Duffy’s Lug Nuts plays on the equivalence people draw between intelligence and arithmetic ability. Also on the idea that brain size should have something particularly strong link to intelligence. Really anyone having trouble figuring out 15% of $10 is psyching themselves out. They’re too much overwhelmed by the idea of percents being complicated to realize that it’s, well, ten times 15 cents.

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


Ah, yes, so, in the midst of feeling all proud that I’d gotten my Reading the Comics workflow improved, I went out to do my afternoon chores without posting the essay. I’m embarrassed. But it really only affects me looking at the WordPress Insights page. It publishes this neat little calendar-style grid that highlights the days when someone’s posted and this breaks up the columns. This can only unnerve me. I deserve it.

Tom Thaves’s Frank and Ernest for the 8th of February is about the struggle to understand zero. As often happens, the joke has a lot of truth to it. Zero bundles together several ideas, overlapping but not precisely equal. And part of that is the idea of “nothing”. Which is a subtly elusive concept: to talk about the properties of a thing that does not exist is hard. As adults it’s easy to not notice this anymore. Part’s likely because mastering a concept makes one forget what it took to understand. Part is likely because if you don’t have to ponder whether the “zero” that’s “one less than one” is the same as the “zero” that denotes “what separates the count of thousands from the count of tens in the numeral 2,038” you might not, and just assume you could explain the difference or similarity to someone who has no idea.

John Zakour and Scott Roberts’s Maria’s Day for the 8th has maria and another girl bonding over their hatred of mathematics. Well, at least they’re getting something out of it. The date in the strip leads me to realize this is probably a rerun. I’m not sure just when it’s from.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th proposes a prank based on mathematical use of the word “arbitrarily”. This is a word that appears a lot in analysis, and the strip makes me realize I’m not sure I can give a precise definition. An “arbitrarily large number”, for example, would be any number that’s large enough. But this also makes me realize I’m not sure precisely what joke Weinersmith is going for. I suppose that if someone were to select an arbitrarily large number they might pick 53, or a hundred, or million billion trillion. I suppose Weinersmith’s point is that in ordinary speech an arbitrarily made choice is one selection from all the possible alternatives. In mathematical speech an arbitrarily made choice reflects every possible choice. To speak of an arbitrarily large number is to say that whatever selection is made, we can go on to show this interesting stuff is true. We’d typically like to prove the most generically true thing possible. But picking a single example can be easier to prove. It can certainly be easier to visualize. 53 is probably easier to imagine than “every number 52 or larger”, for example.

Quincy: 'Someday I'm gonna write a book, Gran.' Grandmom: 'Wonderful. Will you dedicate it to me?' Quincy: 'Sure. In fact, if you want, I'll dedicate this math homework to you.'
Ted Shearer’s Quincy for the 16th of December, 1978 and reprinted the 9th of February, 2018. I’m not sure just what mathematics homework Quincy could be doing to inspire him to write a book, but then, it’s not like my mind doesn’t drift while doing mathematics either. And book-writing’s a common enough daydream that most people are too sensible to act on.

Ted Shearer’s Quincy for the 16th of December, 1978 was rerun the 9th of February. It just shows Quincy at work on his mathematics homework, and considering dedicating it to his grandmother. Mathematics books have dedications, just as any other book does. I’m not aware of dedications of proofs or other shorter mathematics works, but there’s likely some. There’s often a note of thanks, usually given to people who’ve made the paper’s writers think harder about the subjects. But I don’t think there’s any reason a paper wouldn’t thank someone who provided “mere” emotional support. I just don’t have examples offhand.

Jef Mallet’s Frazz for the 9th looks like one of those creative-teaching exercises I sometimes see in Mathematics Education Twitter: the teacher gives answers and the students come up with story problems to match. That’s not a bad project. I’m not sure how to grade it, but I haven’t done anything that creative when I’ve taught. I’m sorry I haven’t got more to say about it since the idea seems fun.

Redeye: 'C'mon, Pokey. Time for your lessons. Okay, what do you get when you divide 5,967,342 by 973 ... ?' Pokey: 'A headache!'
Gordon Bess’s Redeye for the 30th of September, 1971 and reprinted the 10th of February, 2018. I realized I didn’t know the father’s name and looked it up, and Wikipedia revealed to me that he’s named Redeye. You know, like the comic strip implies right there in the title. Look, I just read the comics, I can’t be expected to think about the comics too.

Gordon Bess’s Redeye for the 30th of September, 1971 was rerun the 10th. It’s a bit of extremely long division and I don’t blame Pokey for giving up on that problem. Starting from 5,967,342 divided by 973 I’d say, well, that’s about six million divided by a thousand, so the answer should be near six thousand. I don’t think the last digits of 2 and 3 suggest anything about what the final digit should be, if this divides evenly. So the only guidance I have is that my answer ought to be around six thousand and then we have to go into actually working. It turns out that 973 doesn’t go into 5,967,342 a whole number of times, so I sympathize more with Pokey. The answer is a little more than 6,132.9311.

There’s Technically Still Time To Buy A Theorem For Valentine’s Day


While I have not used the service myself, it does appear that Theory Mine is still going. It’s an automated theorem-creating software. For a price, they’ll create a theorem and name it whatever you choose within reason. It will almost certainly not be an interesting theorem, nor one that anyone will ever care about. But it’ll be far more legitimate than, like, naming a star after someone, which has always been an outright scam.

I discovered this last year, and wrote a bit about how this sort of thing can work. (I’m not certain this is precisely how Theory Mine works, but I am confident that it’s something along these lines.) Also a bit about the history of this sort of system and how it’s come about. And as I say, I haven’t used the service myself. It may sound like bragging, but I’ve created my own theorems. They’re not the monumental triumph of intellect and explanatory power that attaches to theories in science. They can be much more petty things, like “when can we expect the sum of the roots of a quadratic polynomial to be greater than zero”. A theorem you make for your own project will be a little more interesting than a completely auto-generated one like this. After all, your own theorem will at least answer something you wanted to know. A computer-generated one doesn’t even promise that. But it does take less effort to send a bit of money off and get a proof mailed back to you.

Reading the Comics, February 7, 2018: Not Taking Algebra Too Seriously Edition


There were nearly a dozen mathematically-themed comic strips among what I’d read, and they almost but not quite split mid-week. Better, they include one of my favorite ever mathematics strips from Charles Schulz’s Peanuts.

Jimmy Halto’s Little Iodine for the 4th of December, 1956 was rerun the 2nd of February. Little Iodine seeks out help with what seems to be story problems. The rate problem — “if it takes one man two hours to plow seven acros, how long will it take five men and a horse to … ” — is a kind I remember being particularly baffling. I think it’s the presence of three numbers at once. It seems easy to go from, say, “if you go two miles in ten minutes, how long will it take to go six miles?” to an answer. To go from “if one person working two hours plows seven acres then how long will five men take to clear fourteen acres” to an answer seems like a different kind of problem altogether. It’s a kind of problem for which it’s even wiser than usual to carefully list everything you need.

Iodone, going into a department store. 'Boy, we got tough homework for tomorrow.' At Information: 'If it takes one man two hours to plow seven acres, how long will it take five men and a horse to --- etc' Clerk: 'Wha? Uh ... let me get a pencil. Will you repeat that, please? ... Cipher ... two o carry ... mmm ... times x ... minus ... mmm ... now let me think ... ' NEXT DAY; Teacher: 'Sharkey Shannon, 92, very good, Sharkey. Shalimar Shultz, 94, excellent, Shalimar. Iodine Tremblechin ... zero ... every problem wrong! Iodine ... I just can't understand it ... not one single answer correct!' Iodine, at the Complaint Department: 'Somebody in this store has to write a hundred times 'I will henceforth study harder'!
Jimmy Halto’s Little Iodine for the 2nd of December, 1956 and rerun the 4th of February, 2018. It’s the rare Little Iodine where she doesn’t get her father fired!

Kieran Meehan’s Pros and Cons for the 5th uses a bit of arithmetic. It looks as if it’s meant to be a reminder about following the conclusions of one’s deductive logic. It’s more common to use 1 + 1 equalling 2, or 2 + 2 equalling 4. Maybe 2 times 2 being 4. But then it takes a little turn into numerology, trying to read more meaning into numbers than is wise. (I understand why people should use numerological reasoning, especially given how much mathematicians like to talk up mathematics as descriptions of reality and how older numeral systems used letters to represent words. And that before you consider how many numbers have connotations.)

Judge: 'Members of the jury, before retiring to consider your verdict, I shall give you my summing-up. 3 + 3 = 6. There are six letters in the word 'guilty'. Coincidence? I don't believe in coincidences.'
Kieran Meehan’s Pros and Cons for the 5th of February, 2018. I grant the art is a bit less sophisticated than in Little Iodine. But the choice of two features to run outside the panels and into the white gutters is an interesting one and I’m not sure what Meehan is going for in choosing one word balloon and the judge’s hand to run into the space like that.

Charles Schulz’s Peanuts for the 5th of February reruns the strip from the 8th of February, 1971. And it is some of the best advice about finding the values of x and y, and about approaching algebra, that I have ever encountered.

Trixie: 'Look at all the birds!! I wonder how many there are! Sic, nine, five, 'leven, eight, fwee, two! Only two! It sure looked like there were more!'
Mort Walker and Dik Browne’s Hi and Lois for the 10th of August, 1960 was rerun the 6th of February, 2018. And I do like Trixie’s look of bafflement in the last panel there; it’s more expressive than seems usual for the comic even in its 1960s design.

Mort Walker and Dik Browne’s Hi and Lois for the 10th of August, 1960 was rerun the 6th of February. It’s a counting joke. Babies do have some number sense. At least babies as old as Trixie do, I believe, in that they’re able to detect that something weird is going on when they’re shown, eg, two balls put into a box and four balls coming out. (Also it turns out that stage magicians get called in to help psychologists study just how infants and toddlers understand the world, which is neat.)

John Zakour and Scott Roberts’s Maria’s Day for the 7th is Ms Payne’s disappointed attempt at motivating mathematics. I imagine she’d try going on if it weren’t a comic strip limited to two panels.

Reading the Comics, February 3, 2018: Overworked Edition


And this should clear out last week’s mathematically-themed comic strips. I didn’t realize just how busy last week had been until I looked at what I thought was a backlog of just two days’ worth of strips and it turned out to be about two thousand comics. I exaggerate, but as ever, not by much. This current week seems to be a more relaxed pace. So I’ll have to think of something to write for the Tuesday and Thursday slots. Hm. (I’ll be all right. I’ve got one thing I need to stop bluffing about and write, and there’s usually a fair roundup of interesting tweets or articles I’ve seen that I can write. Those are often the most popular articles around here.)

Hilary Price and Rina Piccolo’s Rhymes with Orange for the 1st of February, 2018 gives us an anthropomorphic geometric figures joke for the week. Also a side of these figures that I don’t think I’ve seen in the newspaper comics before. It kind of raises further questions.

The Geometry. A pair of parallel lines, one with a rectangular lump. 'Not true --- parallel lines *do* meet. In fact, Peter and I are expected.' ('We met at a crossroads in both our lives.')
Hilary Price and Rina Piccolo’s Rhymes with Orange for the 1st of February, 2018. All right, but they’re line segments, but I suppose you can’t reasonably draw infinitely vast things in a daily newspaper strip’s space. The lean of that triangle makes it look way more skeptical, even afraid, than I think Price and Piccolo intended, but I’m not sure there’s a better way to get these two in frame without making the composition weird.

Jason Chatfield’s Ginger Meggs for the 1st just mentions that it’s a mathematics test. Ginger isn’t ready for it.

Mark Tatulli’s Heart of the City rerun for the 1st finally has some specific mathematics mentioned in Heart’s efforts to avoid a mathematics tutor. The bit about the sum of adjacent angles forming a right line being 180 degrees is an important one. A great number of proofs rely on it. I can’t deny the bare fact seems dull, though. I know offhand, for example, that this bit about adjacent angles comes in handy in proving that the interior angles of a triangle add up to 180 degrees. At least for Euclidean geometry. And there are non-Euclidean geometries that are interesting and important and for which that’s not true. Which inspires the question: on a non-Euclidean surface, like say the surface of the Earth, is it that adjacent angles don’t add up to 180 degrees? Or does something else in the proof of a triangle’s interior angles adding up to 180 degrees go wrong?

The Eric the Circle rerun for the 2nd, by JohnG, is one of the occasional Erics that talk about π and so get to be considered on-topic here.

Bill Whitehead’s Free Range for the 2nd features the classic page full of equations to demonstrate some hard mathematical work. And it is the sort of subject that is done mathematically. The equations don’t look to me anything like what you’d use for asteroid orbit projections. I’d expect forecasting just where an asteroid might hit the Earth to be done partly by analytic formulas that could be done on a blackboard. And then made precise by a numerical estimate. The advantage of the numerical estimate is that stuff like how air resistance affects the path of something in flight is hard to deal with analytically. Numerically, it’s tedious, but we can let the computer deal with the tedium. So there’d be just a boring old computer screen to show on-panel.

Bud Fisher’s Mutt and Jeff reprint for the 2nd is a little baffling. And not really mathematical. It’s just got a bizarre arithmetic error in it. Mutt’s fiancee Encee wants earrings that cost ten dollars (each?) and Mutt takes this to be fifty dollars in earring costs and I have no idea what happened there. Thomas K Dye, the web cartoonist who’s done artwork for various article series, has pointed out that the lettering on these strips have been redone with a computer font. (Look at the letters ‘S’; once you see it, you’ll also notice it in the slightly lumpy ‘O’ and the curly-arrow ‘G’ shapes.) So maybe in the transcription the earring cost got garbled? And then not a single person reading the finished product read it over and thought about what they were doing? I don’t know.

Zach Weinersmith’s Saturday Morning Breakfast Cereal reprint for the 2nd is based, as his efforts to get my attention often are, on a real mathematical physics postulate. As the woman postulates: given a deterministic universe, with known positions and momentums of every particle, and known forces for how all these interact, it seems like it should be possible to predict the future perfectly. It would also be possible to “retrodict” the past. All the laws of physics that we know are symmetric in time; there’s no reason you can’t predict the motion of something one second into the past just as well as you an one second into the future. This fascinating observation took a lot of battery in the 19th century. Many physical phenomena are better described by statistical laws, particularly in thermodynamics, the flow of heat. In these it’s often possible to predict the future well but retrodict the past not at all.

But that looks as though it’s a matter of computing power. We resort to a statistical understanding of, say, the rings of Saturn because it’s too hard to track the billions of positions and momentums we’d need to otherwise. A sufficiently powerful mathematician, for example God, would be able to do that. Fair enough. Then came the 1890s. Henri Poincaré discovered something terrifying about deterministic systems. It’s possible to have chaos. A mathematical representation of a system is a bit different from the original system. There’s some unavoidable error. That’s bound to make some, larger, error in any prediction of its future. For simple enough systems, this is okay. We can make a projection with an error as small as we need, at the cost of knowing the current state of affairs with enough detail. Poincaré found that some systems can be chaotic, though, ones in which any error between the current system and its representation will grow to make the projection useless. (At least for some starting conditions.) And so many interesting systems are chaotic. Incredibly simplified models of the weather are chaotic; surely the actual thing is. This implies that God’s projection of the universe would be an amusing but almost instantly meaningless toy. At least unless it were a duplicate of the universe. In which case we have to start asking our philosopher friends about the nature of identity and what a universe is, exactly.

Ruben Bolling’s Super-Fun-Pak Comix for the 2nd is an installment of Guy Walks Into A Bar featuring what looks like an arithmetic problem to start. It takes a turn into base-ten jokes. There are times I suspect Ruben Bolling to be a bit of a nerd.

Nate Fakes’s Break of Day for the 3rd looks like it’s trying to be an anthropomorphic-numerals joke. At least it’s an anthropomorphic something joke.

Percy Crosby’s Skippy for the 3rd originally ran the 8th of December, 1930. It alludes to one of those classic probability questions: what’s the chance that in your lungs is one of the molecules exhaled by Julius Caesar in his dying gasp? Or whatever other event you want: the first breath you ever took, or something exhaled by Jesus during the Sermon on the Mount, or exhaled by Sue the T-Rex as she died. Whatever. The chance is always surprisingly high, which reflects the fact there’s a lot of molecules out there. This also reflects a confidence that we can say one molecule of air is “the same” as some molecule if air in a much earlier time. We have to make that supposition to have a problem we can treat mathematically. My understanding is chemists laugh at us if we try to suggest this seriously. Fair enough. But whether the air pumped out of a bicycle tire is ever the same as what’s pumped back in? That’s the same kind of problem. At least some of the molecules of air will be the same ones. Pretend “the same ones” makes sense. Please.

How January 2018 Treated My Mathematics Blog


First of all, I would like to say this about this tweet:

And that is: I don’t feel threatened at all so nyah.

(And if you want to help them out, please, do send any Calvin and Hobbes strips with mathematical themes over their way.)

Back to my usual self-preening. January 2018 was a successful month around here, in terms of people reading stuff I write. According to WordPress, there were some 1,274 pages viewed from 670 unique visitors. That’s the largest number of pages viewed since March and April 2016, when I had a particularly successful A To Z going. It’s the greatest number of unique visitors since September 2017 when I had a less successful but still pretty good A To Z going. The page views were well above December 2017’s 899, and November’s 1,052. The unique visitors were well above December’s 599 and November’s 604.

I don’t have any real explanation for this. I suspect it’s spillover from my humor blog, which had its most popular month since the comic strip Apartment 3-G died a sad, slow, baffling death. Long story. I think my humor blog was popular because people don’t know what happened to the guy who writes Gasoline Alley. I don’t know either, but I tell people if I do find out anything I’ll tell them, and that’s almost as good as knowing something.

Still, this popularity was accompanied by readers actually liking stuff. There were 112 pages liked in January, beating out the 71 in December and 70 in November by literally dozens of clicks. It’s the highest count since August of 2017 and summer’s A To Z sequence. There were more comments, too, 39 of them. December saw 24 and November 28 and, you see this coming, that’s the largest number of comments since summer 2017’s A To Z sequence.

The popular articles for January were two of the ones I expected, one of the Reading the Comics posts, and then two surprises. What were they? These.

Yes, it’s clickbait-y to talk about weird tricks for limits that mathematicians use. In my defense: mathematicians really do rely on these tricks all the time. So if it’s getting people stuff that’s useful then my conscience is as clear as it is for asking “How many grooves are on a record’s side?” and (implicitly) “How many kinds of trapezoid are there?”

If I’m counting right there were 50 countries from which I drew readers, if “European Union” counts as a country and if “Trinidad and Tobago” don’t count as two. Plus there’s Hong Kong and when you get down to it, “country” is a hard concept to pin down exactly. There were 14 single-reader countries. Here’s the roster of them all:

Country Readers
United States 879
India 89
Philippines 59
United Kingdom 37
Canada 28
Singapore 15
Hong Kong SAR China 11
Netherlands 11
Sweden 11
Belgium 9
Algeria 8
Austria 8
Australia 7
France 7
Italy 7
Switzerland 7
South Africa 6
Brazil 5
Slovenia 5
Argentina 4
Germany 4
Japan 4
Pakistan 4
Indonesia 3
Spain 3
Denmark 2
Egypt 2
European Union 2
Greece 2
Iraq 2
New Zealand 2
Portugal 2
South Korea 2
Thailand 2
Ukraine 2
Bulgaria 1
Czech Republic 1
Ireland 1
Malaysia 1
Mexico 1 (**)
Namibia 1
Norway 1
Russia 1 (*)
Saudi Arabia 1
Sri Lanka 1
Trinidad & Tobago 1
Turkey 1
Uruguay 1 (*)
Vietnam 1

There were 53 countries sending me readers in December and 56 in November so I guess I’m concentrating? There were 15 single-reader countries in December and 22 in November. Russia and Uruguay were single-reader countries in December; Mexico’s been a single-reader country for three months now.

WordPress’s Insights panel says I started the month with 57,592 page views recorded, from 27,161 recorded unique visitors. It also shares with me the interesting statistics that, as I write this and before I post it, I’ve written 16 total posts this year, which have drawn an average two comments and seven likes per post. There’ve been 900 words per post, on average. Overall this year I’ve gotten 39 comments, 110 likes, and have published 14,398 words. I don’t know whether that counts image captions. But this also leads me to learn what previous year statistics were like; I’ve been averaging over 900 words per post since 2015. In 2015 I averaged about 750 words per post, and got three times as many likes and about twice as many comments per post. I’m sure that doesn’t teach me anything. At the least I won’t learn from it.

If all this has convinced you to read my posts, please, keep reading them. You can add them to a WordPress reader by way of the “Follow nebusresearch” sticker on the center-right of the page. Or you can get it delivered by e-mail using the “Follow Blog Via E-Mail” button underneath it. If you’ve got your own RSS reader, you can follow from this feed. There’s probably more ways to follow this, too. And if you want to follow me on Twitter, try @Nebusj, because that’s me and I like having company there.

Reading the Comics, January 31, 2018: Workload Edition


I thought my new workflow of writing my paragraph or two about each comic was going to help me keep up and keep fresher with the daily comics. And then Comic Strip Master Command decided that everybody had to do comics that at least touched on some mathematical subject. I don’t know. I’m trying to keep up but will admit, I didn’t get to writing anything about Friday’s or Saturday’s strips yet. They’ll keep a couple days.

Bill Amend’s FoxTrot Classicsfor the 29th of January reprints the strip from the 5th of February, 1996. (The Classics reprints finally reached the point where Amend retired from daily strips, and jumped back a dozen years to continue printing.) It just mentions mathematics exams, and high performances on both is all.

Josh Shalek’s Kid Shay Comics reprint for the 29th tosses off a mention of Uncle Brian attempting a great mathematical feat. In this case it’s the Grand Unification Theory, some logically coherent set of equations that describe the fundamental forces of the universe. I think anyone with a love for mathematics makes a couple quixotic attempts on enormously vast problems like this. Or the Riemann Hypothesis, or Goldbach’s Conjecture, or Fermat’s Last Theorem. Yes, Fermat’s Last Theorem has been proven, but there’s no reason there couldn’t be an easier proof. Similarly there’s no reason there couldn’t be a better proof of the Four Color Map theorem. Most of these attempts end up the way Brian’s did. But there’s value in attempting this anyway. Even when you fail, you can have fun and learn fascinating things in the attempt.

Carol Lay’s Lay Lines for the 29th is a vignette about a statistician. And one of those statisticians with the job of finding surprising correlations between things. I think it’s also a riff on the hypothesis that free markets are necessarily perfect: if there’s any advantage to doing something one way, it’ll quickly be found and copied until that is the normal performance of the market. Anyone doing better than average is either taking advantage of concealed information, or else is lucky.

Matt Lubchansky’s Please Listen To Me for the 29th depicts a person doing statistical work for his own purposes. In this case he’s trying to find what factors might be screwing up the world. The expressions in the second panel don’t have an obvious meaning to me. The start of the expression \int exp\left(\frac{1}{N_0}\right) at the top line suggests statistical mechanics to me, for what that’s worth, and the H and Ψ underneath suggest thermodynamics or quantum mechanics. So if Lubchansky was just making up stuff, he was doing it with a good eye for mathematics that might underly everything.

Rick Stromoski’s Soup to Nutz for the 29th circles around the anthropomorphic numerals idea. It’s not there exactly, but Andrew is spending some time giving personality to numerals. I can’t say I give numbers this much character. But there are numbers that seem nicer than others. Usually this relates to what I can do with the numbers. 10, for example, is so easy to multiply or divide by. If I need to multiply a number by, say, something near thirty, it’s a delight to triple it and then multiply by ten. Twelve and 24 and 60 are fun because they’re so relatively easy to find parts of. Even numbers often do seem easier to work with, just because splitting an even number in half saves us from dealing with decimals or fractions. Royboy sees all this as silliness, which seems out of character for him, really. I’d expect him to be up for assigning traits to numbers like that.

Zippy, in front of the Wein-O-Rama Restaurant: 'Different parts of Einstein's theory of relativity are being proven true all th'time ... hmm ... what about th'idea that an exact DUPLICATE of everything exists at th'same time on th'other side of th'universe? Or, in this case, on 'th'other side of th'the Wein-O-Rama restaurant!' Other Zippy, at the other end of the the Wein-O-Rama restaurant: 'What happens in Rhode Island stays in Rhode Island! Yow!'
Bill Griffith’s Zippy the Pinhead for the 30th of January, 2018. The the Wein-O-Rama is in Cranston, Rhode Island, and I do hope that this strip is now part of the “In Pop Culture” segment of Cranson’s Wikipedia page. DuckDuckGo’s search for Wein-O-Rama pops up, for me, this sentence from a TripAdvisor.com review: “When we go to Weinorama its [sic] always for the wieners [sic], that `RI-only’ treat. I’ll always order them all the way, but my wife always says ‘no onions’.” Thus does reality merge imperceptibly into Zippy the Pinhead comics, evoking the surrealist character’s ancient dictum, “Life is a blur of Republicans and meat”.

Bill Griffith’s Zippy the Pinhead for the 30th mentions Albert Einstein and relativity. And Zippy ruminates on the idea that there’s duplicates of everything, in the vastness of the universe. It’s an unsettling idea that isn’t obviously ruled out by mathematics alone. There’s, presumably, some chance that a bunch of carbon and hydrogen and oxygen and other atoms happened to come together in such a way as to make our world as we know it today. If there’s a vast enough universe, isn’t there a chance that a bunch of carbon and hydrogen and oxygen and other atoms happened to come together that same way twice? Three times? If the universe is infinitely large, might it not happen infinitely many times? In any number of variations? It’s hard to see why not, but even if it is possible, that’s no reason to think it must happen either. And whether those duplicates are us is a question for philosophers studying the problem of identity and what it means to be one person rather than some other person. (It turns out to be a very difficult problem and I’m glad I’m not expected to offer answers.)

Tony Cochrane’s Agnes attempts to use mathematics to reason her way to a better bedtime the 31st. She’s not doing well. Also this seems like it’s more of an optimization problem than a simple arithmetic one. What’s the latest bedtime she can get that still allows for everything that has to be done, likely including getting up in time and getting enough sleep? Also, just my experience but I didn’t think Agnes was old enough to stay up until 10 in the first place.

Reading the Comics, January 27, 2018: Working Through The Week Edition


And today I bring the last couple mathematically-themed comic strips sent my way last week. GoComics has had my comics page working intermittently this week. And I was able to get a response from them, by e-mailing their international sales office, the only non-form contact I could find. Anyway, this flood of comics does take up the publishing spot I’d figured for figuring how I messed up Wronski’s formula. But that’s all right, as I wanted to spend more time thinking about that. Here’s hoping spending more time thinking works out for me.

Nate Fakes’s Break of Day for the 24th was the big anthropomorphic numerals joke for the week. And it’s even dubbed the numbers game.

Mark Tatulli’s Heart of the City from the 24th got into a storyline about Heart needing a mathematics tutor. It’s a rerun sequence, although if you remember a particular comic storyline from 2009 you’re doing pretty well. Nothing significantly mathematical has turned up in the story so far, past the mention of fractions as things that exist and torment students. But the stories are usually pretty good for this sort of strip.

Mikael Wulff and Anders Morganthaler’s WuMo for the 24th includes a story problems freak out. I’m not sure what’s particularly implausible about buying nine apples. I’d agree a person is probably more likely to buy an even number of things, since we seem to like numbers like “ten” and “eight” so well, but it’s hardly ridiculous.

Tim Rickard’s Brewster Rockit for the 25th is an arithmetic class on the Snowman Planet. So there’s some finger-counting involved.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th is a reminder that most of my days are spent seeing how Zach Weinersmith wants my attention. It also includes what I suppose is a legitimate attempt to offer a definition for what all mathematics is. It’s hard to come up with something that does cover all the stuff mathematicians do. Bear in mind, this includes counting, calculating how far the Sun is based on the appearance of a lunar eclipse, removing static from a recording, and telling how many queens it’s possible to place eight queens on a chess board that’s wrapped around a torus without any being able to capture another, among other problems. My instinct is to dismiss the proposed “anything you can think deeply about that has no reference to the real world”. That seems over-broad, and to cover a lot of areas that are really philosophy’s beat. And I think there’s something unseemly in mathematicians gloating about their work having no “practical” use. I grant I come from an applied school, and I came to there through an interest in physics. But to build up “inapplicability to the real word” as if it were some ideal, as opposed to just how something has turned out to be right now, strikes me as silly. Applicability is so dependent on context, on culture, and accidents of fate that there’s no way it can be important to characterizing mathematics. And it would imply that once we found a use for something it would stop being mathematically interesting. I don’t see evidence of that in mathematical history.

Mikael Wulff and Anders Morganthaler’s WuMo pops back in on the 27th with an appearance of sudoku, presenting the logic puzzle as one of the many things beyond the future Disgraced Former President’s abilities.

Reading the Comics, January 23, 2018: Adult Content Edition


I was all set to say how complaining about GoComics.com’s pages not loading had gotten them fixed. But they only worked for Monday alone; today they’re broken again. Right. I haven’t tried sending an error report again; we’ll see if that works. Meanwhile, I’m still not through last week’s comic strips and I had just enough for one day to nearly enough justify an installment for the one day. Should finish off the rest of the week next essay, probably in time for next week.

Mark Leiknes’s Cow and Boy rerun for the 23rd circles around some of Zeno’s Paradoxes. At the heart of some of them is the question of whether a thing can be divided infinitely many times, or whether there must be some smallest amount of a thing. Zeno wonders about space and time, but you can do as well with substance, with matter. Mathematics majors like to say the problem is easy; Zeno just didn’t realize that a sum of infinitely many things could be a finite and nonzero number. This misses the good question of how the sum of infinitely many things, none of which are zero, can be anything but infinitely large? Or, put another way, what’s different in adding \frac11 + \frac12 + \frac13 + \frac14 + \cdots and adding \frac11 + \frac14 + \frac19 + \frac{1}{16} + \cdots that the one is infinitely large and the other not?

Or how about this. Pick your favorite string of digits. 23. 314. 271828. Whatever. Add together the series \frac11 + \frac12 + \frac13 + \frac14 + \cdots except that you omit any terms that have your favorite string there. So, if you picked 23, don’t add \frac{1}{23} , or \frac{1}{123} , or \frac{1}{802301} or such. That depleted series does converge. The heck is happening there? (Here’s why it’s true for a single digit being thrown out. Showing it’s true for longer strings of digits takes more work but not really different work.)

J C Duffy’s Lug Nuts for the 23rd is, I think, the first time I have to give a content warning for one of these. It’s a porn-movie advertisement spoof. But it mentions Einstein and Pi and has the tagline “she didn’t go for eggheads … until he showed her a new equation!”. So, you know, it’s using mathematics skill as a signifier of intelligence and riffing on the idea that nerds like sex too.

John Graziano’s Ripley’s Believe It or Not for the 23rd has a trivia that made me initially think “not”. It notes Vince Parker, Senior and Junior, of Alabama were both born on Leap Day, the 29th of February. I’ll accept this without further proof because of the very slight harm that would befall me were I to accept this wrongly. But it also asserted this was a 1-in-2.1-million chance. That sounded wrong. Whether it is depends on what you think the chance is of.

Because what’s the remarkable thing here? That a father and son have the same birthday? Surely the chance of that is 1 in 365. The father could be born any day of the year; the son, also any day. Trusting there’s no influence of the father’s birthday on the son’s, then, 1 in 365 it is. Or, well, 1 in about 365.25, since there are leap days. There’s approximately one leap day every four years, so, surely that, right?

And not quite. In four years there’ll be 1,461 days. Four of them will be the 29th of January and four the 29th of September and four the 29th of August and so on. So if the father was born any day but leap day (a “non-bissextile day”, if you want to use a word that starts a good fight in a Scrabble match), the chance the son’s birth is the same is 4 chances in 1,461. 1 in 365.25. If the father was born on Leap Day, then the chance the son was born the same day is only 1 chance in 1,461. Still way short of 1-in-2.1-million. So, Graziano’s Ripley’s is wrong if that’s the chance we’re looking at.

Ah, but what if we’re looking at a different chance? What if we’re looking for the chance that the father is born the 29th of February and the son is also born the 29th of February? There’s a 1-in-1,461 chance the father’s born on Leap Day. And a 1-in-1,461 chance the son’s born on Leap Day. And if those events are independent, the father’s birth date not influencing the son’s, then the chance of both those together is indeed 1 in 2,134,521. So Graziano’s Ripley’s is right if that’s the chance we’re looking at.

Which is a good reminder: if you want to work out the probability of some event, work out precisely what the event is. Ordinary language is ambiguous. This is usually a good thing. But it’s fatal to discussing probability questions sensibly.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 23rd presents his mathematician discovering a new set of numbers. This will happen. Mathematics has had great success, historically, finding new sets of things that look only a bit like numbers were understood. And showing that if they follow rules that are, as much as possible, like the old numbers, we get useful stuff out of them. The mathematician claims to be a formalist, in the punch line. This is a philosophy that considers mathematical results to be the things you get by starting with some symbols and some rules for manipulating them. What this stuff means, and whether it reflects anything of interest in the real world, isn’t of interest. We can know the results are good because they follow the rules.

This sort of approach can be fruitful. It can force you to accept results that are true but intuition-defying. And it can give results impressive confidence. You can even, at least in principle, automate the creating and the checking of logical proofs. The disadvantages are that it takes forever to get anything done. And it’s hard to shake the idea that we ought to have some idea what any of this stuff means.

Reading the Comics, January 22, 2018: Breaking Workflow Edition


So I was travelling last week, and this threw nearly all my plans out of whack. We stayed at one of those hotels that’s good enough that its free Internet is garbage and they charge you by day for decent Internet. So naturally Comic Strip Master Command sent a flood of posts. I’m trying to keep up and we’ll see if I wrap up this past week in under three essays. And I am not helped, by the way, by GoComics.com rejiggering something on their server so that My Comics Page won’t load, and breaking their “Contact Us” page so that that won’t submit error reports. If someone around there can break in and turn one of their servers off and on again, I’d appreciate the help.

Hy Eisman’s Katzenjammer Kids for the 21st of January is a curiously-timed Tax Day joke. (Well, the Katzenjammer Kids lapsed into reruns a dozen years ago and there’s probably not much effort being put into selecting seasonally appropriate ones.) But it is about one of the oldest and still most important uses of mathematics, and one that never gets respect.

Mama: 'Der deadline fer der kink's taxes iss dis veek! Der kink's new tax law makes gif'ink him yer money much easier!' Captain: 'Mit der new forms it should be a snep!' All that day ... Captain: 'Let's see. Add lines 4, 8 und 12 to line 18 und subtract line 22'. And also the next day. Captain: 'Add der number uf fish caught by you diss year und divide by der veight uf der bait ...' And the day after that ... 'If you ate t'ree meals a day all t'rough der year, check idss box ... if you vun money playink pinochle mit der Kink, enter der amount ... ' As the Captain throws the forms up, Mama says, 'Captain! Der tax collector iss here!' The Captain raspberries the agent: 'Hey! Tax collector!' Next panel, in prison. Mama: 'Dumkopf! Why din't you fill out der new easy tax forms?' Captain, in chains: 'Diss iss easier!'
Hy Eisman’s Katzenjammer Kids for the 21st of January, 2018. And, fine, but if the tax forms are that impossible to do right then shouldn’t there be a lot more people in jail for the same problem? … Although I suppose the comic strip hasn’t got enough of a cast for that.

Morrie Turner’s Wee Pals rerun for the 21st gets Oliver the reputation for being a little computer because he’s good at arithmetic. There is something that amazes in a person who’s able to calculate like this without writing anything down or using a device to help.

Steve Kelley and Jeff Parker’s Dustin for the 22nd seems to be starting off with a story problem. It might be a logic problem rather than arithmetic. It’s hard to say from what’s given.

Dustin: 'Next problem. Howard mails letters to four friends: Don, Mary, Tom, and Liz. It takes two days for the letter to get to Don.' Student: 'Excuse me? What's a letter?' Other student: 'Dude, it's the paper the mailman brings for your parents to put in the recycling.'
Steve Kelley and Jeff Parker’s Dustin for the 22nd of January, 2018. Yeah, yeah, people don’t send letters anymore and there’s an eternal struggle to make sure that story problems track with stuff that the students actually do, or know anything about. I still feel weird about how often the comic approaches Ruben Bolling’s satirical Comics For The Elderly. Usually Dustin (the teacher here) is getting the short end; it’s odd that he isn’t, for a change.

Mark Anderson’s Andertoons for the 22nd is the Mark Anderson’s Andertoons for the week. Well, for Monday, as I write this. It’s got your classic blackboard full of equations for the people in over their head. The equations look to me like gibberish. There’s a couple diagrams of aromatic organic compounds, which suggests some quantum-mechanics chemistry problem, if you want to suppose this could be narrowed down.

Greg Evans’s Luann Againn for the 22nd has Luann despair about ever understanding algebra without starting over from scratch and putting in excessively many hours of work. Sometimes it feels like that. My experience when lost in a subject has been that going back to the start often helps. It can be easier to see why a term or a concept or a process is introduced when you’ve seen it used some, and often getting one idea straight will cause others to fall into place. When that doesn’t work, trying a different book on the same topic — even one as well-worn as high school algebra — sometimes helps. Just a different writer, or a different perspective on what’s key, can be what’s needed. And sometimes it just does take time working at it all.

Richard Thompson’s Richard’s Poor Almanac rerun for the 22nd includes as part of a kit of William Shakespeare paper dolls the Typing Monkey. It’s that lovely, whimsical figure that might, in time, produce any written work you could imagine. I think I’d retired monkeys-at-typewriters as a thing to talk about, but I’m easily swayed by Thompson’s art and comic stylings so here it is.

Darrin Bell and Theron Heir’s Rudy Park for the 18th throws around a lot of percentages. It’s circling around the sabermetric-style idea that everything can be quantified, and measured, and that its changes can be tracked. In this case it’s comments on Star Trek: Discovery, but it could be anything. I’m inclined to believe that yeah, there’s an astounding variety of things that can be quantified and measured and tracked. But it’s also easy, especially when you haven’t got a good track record of knowing what is important to measure, to start tracking what amounts to random noise. (See any of my monthly statistics reviews, when I go looking into things like views-per-visitor-per-post-made or some other dubiously meaningful quantity.) So I’m inclined to side with Randy and his doubts that the Math Gods sanction this much data-mining.

Wronski’s Formula For Pi: How Close We Came


Previously:


Józef Maria Hoëne-Wronski’s had an idea for a new, universal, culturally-independent definition of π. It was this formula that nobody went along with because they had looked at it:

\pi = \frac{4\infty}{\sqrt{-1}}\left\{ \left(1 + \sqrt{-1}\right)^{\frac{1}{\infty}} -  \left(1 - \sqrt{-1}\right)^{\frac{1}{\infty}} \right\}

I made some guesses about what he would want this to mean. And how we might put that in terms of modern, conventional mathematics. I describe those in the above links. In terms of limits of functions, I got this:

\displaystyle  \lim_{x \to \infty} f(x) = \lim_{x \to \infty} -2 x 2^{\frac{1}{2}\cdot \frac{1}{x}} \sin\left(\frac{\pi}{4}\cdot \frac{1}{x}\right)

The trouble is that limit took more work than I wanted to do to evaluate. If you try evaluating that ‘f(x)’ at ∞, you get an expression that looks like zero times ∞. This begs for the use of L’Hôpital’s Rule, which tells you how to find the limit for something that looks like zero divided by zero, or like ∞ divided by ∞. Do a little rewriting — replacing that first ‘x’ with ‘\frac{1}{1 / x} — and this ‘f(x)’ behaves like L’Hôpital’s Rule needs.

The trouble is, that’s a pain to evaluate. L’Hôpital’s Rule works on functions that look like one function divided by another function. It does this by calculating the derivative of the numerator function divided by the derivative of the denominator function. And I decided that was more work than I wanted to do.

Where trouble comes up is all those parts where \frac{1}{x} turns up. The derivatives of functions with a lot of \frac{1}{x} terms in them get more complicated than the original functions were. Is there a way to get rid of some or all of those?

And there is. Do a change of variables. Let me summon the variable ‘y’, whose value is exactly \frac{1}{x} . And then I’ll define a new function, ‘g(y)’, whose value is whatever ‘f’ would be at \frac{1}{y} . That is, and this is just a little bit of algebra:

g(y) = -2 \cdot \frac{1}{y} \cdot 2^{\frac{1}{2} y } \cdot \sin\left(\frac{\pi}{4} y\right)

The limit of ‘f(x)’ for ‘x’ at ∞ should be the same number as the limit of ‘g(y)’ for ‘y’ at … you’d really like it to be zero. If ‘x’ is incredibly huge, then \frac{1}{x} has to be incredibly small. But we can’t just swap the limit of ‘x’ at ∞ for the limit of ‘y’ at 0. The limit of a function at a point reflects the value of the function at a neighborhood around that point. If the point’s 0, this includes positive and negative numbers. But looking for the limit at ∞ gets at only positive numbers. You see the difference?

… For this particular problem it doesn’t matter. But it might. Mathematicians handle this by taking a “one-sided limit”, or a “directional limit”. The normal limit at 0 of ‘g(y)’ is based on what ‘g(y)’ looks like in a neighborhood of 0, positive and negative numbers. In the one-sided limit, we just look at a neighborhood of 0 that’s all values greater than 0, or less than 0. In this case, I want the neighborhood that’s all values greater than 0. And we write that by adding a little + in superscript to the limit. For the other side, the neighborhood less than 0, we add a little – in superscript. So I want to evalute:

\displaystyle  \lim_{y \to 0^+} g(y) = \lim_{y \to 0^+}  -2\cdot\frac{2^{\frac{1}{2}y} \cdot \sin\left(\frac{\pi}{4} y\right)}{y}

Limits and L’Hôpital’s Rule and stuff work for one-sided limits the way they do for regular limits. So there’s that mercy. The first attempt at this limit, seeing what ‘g(y)’ is if ‘y’ happens to be 0, gives -2 \cdot \frac{1 \cdot 0}{0} . A zero divided by a zero is promising. That’s not defined, no, but it’s exactly the format that L’Hôpital’s Rule likes. The numerator is:

-2 \cdot 2^{\frac{1}{2}y} \sin\left(\frac{\pi}{4} y\right)

And the denominator is:

y

The first derivative of the denominator is blessedly easy: the derivative of y, with respect to y, is 1. The derivative of the numerator is a little harder. It demands the use of the Product Rule and the Chain Rule, just as last time. But these chains are easier.

The first derivative of the numerator is going to be:

-2 \cdot 2^{\frac{1}{2}y} \cdot \log(2) \cdot \frac{1}{2} \cdot \sin\left(\frac{\pi}{4} y\right) + -2 \cdot 2^{\frac{1}{2}y} \cdot \cos\left(\frac{\pi}{4} y\right) \cdot \frac{\pi}{4}

Yeah, this is the simpler version of the thing I was trying to figure out last time. Because this is what’s left if I write the derivative of the numerator over the derivative of the denominator:

\displaystyle  \lim_{y \to 0^+} \frac{ -2 \cdot 2^{\frac{1}{2}y} \cdot \log(2) \cdot \frac{1}{2} \cdot \sin\left(\frac{\pi}{4} y\right) + -2 \cdot 2^{\frac{1}{2}y} \cdot \cos\left(\frac{\pi}{4} y\right) \cdot \frac{\pi}{4} }{1}

And now this is easy. Promise. There’s no expressions of ‘y’ divided by other expressions of ‘y’ or anything else tricky like that. There’s just a bunch of ordinary functions, all of them defined for when ‘y’ is zero. If this limit exists, it’s got to be equal to:

\displaystyle  -2 \cdot 2^{\frac{1}{2} 0} \cdot \log(2) \cdot \frac{1}{2} \cdot \sin\left(\frac{\pi}{4} \cdot 0\right) + -2 \cdot 2^{\frac{1}{2} 0 } \cdot \cos\left(\frac{\pi}{4} \cdot 0\right) \cdot \frac{\pi}{4}

\frac{\pi}{4} \cdot 0 is 0. And the sine of 0 is 0. The cosine of 0 is 1. So all this gets to be a lot simpler, really fast.

\displaystyle  -2 \cdot 2^{0} \cdot \log(2) \cdot \frac{1}{2} \cdot 0 + -2 \cdot 2^{ 0 } \cdot 1 \cdot \frac{\pi}{4}

And 20 is equal to 1. So the part to the left of the + sign there is all zero. What remains is:

\displaystyle   0 + -2 \cdot \frac{\pi}{4}

And so, finally, we have it. Wronski’s formula, as best I make it out, is a function whose value is …

-\frac{\pi}{2}

… So, what Wronski had been looking for, originally, was π. This is … oh, so very close to right. I mean, there’s π right there, it’s just multiplied by an unwanted -\frac{1}{2} . The question is, where’s the mistake? Was Wronski wrong to start with? Did I parse him wrongly? Is it possible that the book I copied Wronski’s formula from made a mistake?

Could be any of them. I’d particularly suspect I parsed him wrongly. I returned the library book I had got the original claim from, and I can’t find it again before this is set to publish. But I should check whether Wronski was thinking to find π, the ratio of the circumference to the diameter of a circle. Or might he have looked to find the ratio of the circumference to the radius of a circle? Either is an interesting number worth finding. We’ve settled on the circumference-over-diameter as valuable, likely for practical reasons. It’s much easier to measure the diameter than the radius of a thing. (Yes, I have read the Tau Manifesto. No, I am not impressed by it.) But if you know 2π, then you know π, or vice-versa.

The next question: yeah, but I turned up -½π. What am I talking about 2π for? And the answer there is, I’m not the first person to try working out Wronski’s stuff. You can try putting the expression, as best you parse it, into a tool like Mathematica and see what makes sense. Or you can read, for example, Quora commenters giving answers with way less exposition than I do. And I’m convinced: somewhere along the line I messed up. Not in an important way, but, essentially, doing something equivalent to divided by -2 when I should have multiplied by that.

I’ve spotted my mistake. I figure to come back around to explaining where it is and how I made it.

Reading the Comics, January 20, 2018: Increased Workload Edition


It wasn’t much of an increased workload, really. I mean, none of the comics required that much explanation. But Comic Strip Master Command donated enough topics to me last week that I have a second essay for the week. And here it is; sorry there’s no pictures.

Mark Anderson’s Andertoons for the 17th is the Mark Anderson’s Andertoons we’ve been waiting for. It returns to fractions and their frustrations for its comic point.

Jef Mallet’s Frazz for the 17th talks about story problems, although not to the extent of actually giving one as an example. It’s more about motivating word-problem work.

Mike Thompson’s Grand Avenue for the 17th is an algebra joke. I’d call it a cousin to the joke about mathematics’s ‘x’ not coming back and we can’t say ‘y’. On the 18th was one mentioning mathematics, although in a joke structure that could have been any subject.

Lorrie Ransom’s The Daily Drawing for the 18th is another name-drop of mathematics. I guess it’s easier to use mathematics as the frame for saying something’s just a “problem”. I don’t think of, say, identifying the themes of a story as a problem in the way that finding the roots of a quadratic is.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 18th is an anthropomorphic-geometric-figures joke that I’m all but sure is a rerun I’ve shared here before. I’ll try to remember to check before posting this.

Mikael Wulff and Anders Morgenthaler’s WuMo for the 20th gives us a return of the pie chart joke that seems like it’s been absent a while. Worth including? Eh, why not.