As I Try To Figure Out What Wronski Thought ‘Pi’ Was

A couple weeks ago I shared a fascinating formula for π. I got it from Carl B Boyer’s The History of Calculus and its Conceptual Development. He got it from Józef Maria Hoëne-Wronski, early 19th-century Polish mathematician. His idea was that an absolute, culturally-independent definition of π would come not from thinking about circles and diameters but rather this formula:

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

Now, this formula is beautiful, at least to my eyes. It’s also gibberish. At least it’s ungrammatical. Mathematicians don’t like to write stuff like “four times infinity”, at least not as more than a rough draft on the way to a real thought. What does it mean to multiply four by infinity? Is arithmetic even a thing that can be done on infinitely large quantities? Among Wronski’s problems is that they didn’t have a clear answer to this. We’re a little more advanced in our mathematics now. We’ve had a century and a half of rather sound treatment of infinitely large and infinitely small things. Can we save Wronski’s work?

Start with the easiest thing. I’m offended by those $\sqrt{-1}$ bits. Well, no, I’m more unsettled by them. I would rather have $\imath$ in there. The difference? … More taste than anything sound. I prefer, if I can get away with it, using the square root symbol to mean the positive square root of the thing inside. There is no positive square root of -1, so, pfaugh, away with it. Mere style? All right, well, how do you know whether those $\sqrt{-1}$ terms are meant to be $\imath$ or its additive inverse, $-\imath$? How do you know they’re all meant to be the same one? See? … As with all style preferences, it’s impossible to be perfectly consistent. I’m sure there are times I accept a big square root symbol over a negative or a complex-valued quantity. But I’m not forced to have it here so I’d rather not. First step:

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

Also dividing by $\imath$ is the same as multiplying by $-\imath$ so the second easy step gives me:

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

Now the hard part. All those infinities. I don’t like multiplying by infinity. I don’t like dividing by infinity. I really, really don’t like raising a quantity to the one-over-infinity power. Most mathematicians don’t. We have a tool for dealing with this sort of thing. It’s called a “limit”.

Mathematicians developed the idea of limits over … well, since they started doing mathematics. In the 19th century limits got sound enough that we still trust the idea. Here’s the rough way it works. Suppose we have a function which I’m going to name ‘f’ because I have better things to do than give functions good names. Its domain is the real numbers. Its range is the real numbers. (We can define functions for other domains and ranges, too. Those definitions look like what they do here.)

I’m going to use ‘x’ for the independent variable. It’s any number in the domain. I’m going to use ‘a’ for some point. We want to know the limit of the function “at a”. ‘a’ might be in the domain. But — and this is genius — it doesn’t have to be. We can talk sensibly about the limit of a function at some point where the function doesn’t exist. We can say “the limit of f at a is the number L”. I hadn’t introduced ‘L’ into evidence before, but … it’s a number. It has some specific set value. Can’t say which one without knowing what ‘f’ is and what its domain is and what ‘a’ is. But I know this about it.

Pick any error margin that you like. Call it ε because mathematicians do. However small this (positive) number is, there’s at least one neighborhood in the domain of ‘f’ that surrounds ‘a’. Check every point in that neighborhood other than ‘a’. The value of ‘f’ at all those points in that neighborhood other than ‘a’ will be larger than L – ε and smaller than L + ε.

Yeah, pause a bit there. It’s a tricky definition. It’s a nice common place to crash hard in freshman calculus. Also again in Intro to Real Analysis. It’s not just you. Perhaps it’ll help to think of it as a kind of mutual challenge game. Try this.

1. You draw whatever error bar, as big or as little as you like, around ‘L’.
2. But I always respond by drawing some strip around ‘a’.
3. You then pick absolutely any ‘x’ inside my strip, other than ‘a’.
4. Is f(x) always within the error bar you drew?

Suppose f(x) is. Suppose that you can pick any error bar however tiny, and I can answer with a strip however tiny, and every single ‘x’ inside my strip has an f(x) within your error bar … then, L is the limit of f at a.

Again, yes, tricky. But mathematicians haven’t found a better definition that doesn’t break something mathematicians need.

To write “the limit of f at a is L” we use the notation:

$\displaystyle \lim_{x \to a} f(x) = L$

The ‘lim’ part probably makes perfect sense. And you can see where ‘f’ and ‘a’ have to enter into it. ‘x’ here is a “dummy variable”. It’s the falsework of the mathematical expression. We need some name for the independent variable. It’s clumsy to do without. But it doesn’t matter what the name is. It’ll never appear in the answer. If it does then the work went wrong somewhere.

What I want to do, then, is turn all those appearances of ‘∞’ in Wronski’s expression into limits of something at infinity. And having just said what a limit is I have to do a patch job. In that talk about the limit at ‘a’ I talked about a neighborhood containing ‘a’. What’s it mean to have a neighborhood “containing ∞”?

The answer is exactly what you’d think if you got this question and were eight years old. The “neighborhood of infinity” is “all the big enough numbers”. To make it rigorous, it’s “all the numbers bigger than some finite number that let’s just call N”. So you give me an error bar around ‘L’. I’ll give you back some number ‘N’. Every ‘x’ that’s bigger than ‘N’ has f(x) inside your error bars. And note that I don’t have to say what ‘f(∞)’ is or even commit to the idea that such a thing can be meaningful. I only ever have to think directly about values of ‘f(x)’ where ‘x’ is some real number.

So! First, let me rewrite Wronski’s formula as a function, defined on the real numbers. Then I can replace each ∞ with the limit of something at infinity and … oh, wait a minute. There’s three ∞ symbols there. Do I need three limits?

Ugh. Yeah. Probably. This can be all right. We can do multiple limits. This can be well-defined. It can also be a right pain. The challenge-and-response game needs a little modifying to work. You still draw error bars. But I have to draw multiple strips. One for each of the variables. And every combination of values inside all those strips has give an ‘f’ that’s inside your error bars. There’s room for great mischief. You can arrange combinations of variables that look likely to break ‘f’ outside the error bars.

So. Three independent variables, all taking a limit at ∞? That’s not guaranteed to be trouble, but I’d expect trouble. At least I’d expect something to keep the limit from existing. That is, we could find there’s no number ‘L’ so that this drawing-neighborhoods thing works for all three variables at once.

Let’s try. One of the ∞ will be a limit of a variable named ‘x’. One of them a variable named ‘y’. One of them a variable named ‘z’. Then:

$f(x, y, z) = -4 \imath x \left\{ \left(1 + \imath\right)^{\frac{1}{y}} - \left(1 - \imath\right)^{\frac{1}{z}} \right\}$

Without doing the work, my hunch is: this is utter madness. I expect it’s probably possible to make this function take on many wildly different values by the judicious choice of ‘x’, ‘y’, and ‘z’. Particularly ‘y’ and ‘z’. You maybe see it already. If you don’t, you maybe see it now that I’ve said you maybe see it. If you don’t, I’ll get there, but not in this essay. But let’s suppose that it’s possible to make f(x, y, z) take on wildly different values like I’m getting at. This implies that there’s not any limit ‘L’, and therefore Wronski’s work is just wrong.

Thing is, Wronski wouldn’t have thought that. Deep down, I am certain, he thought the three appearances of ∞ were the same “value”. And that to translate him fairly we’d use the same name for all three appearances. So I am going to do that. I shall use ‘x’ as my variable name, and replace all three appearances of ∞ with the same variable and a common limit. So this gives me the single function:

$f(x) = -4 \imath x \left\{ \left(1 + \imath\right)^{\frac{1}{x}} - \left(1 - \imath\right)^{\frac{1}{x}} \right\}$

And then I need to take the limit of this at ∞. If Wronski is right, and if I’ve translated him fairly, it’s going to be π. Or something easy to get π from.

I hope to get there next week.

When Is Thanksgiving Most Likely To Happen?

I thought I had written this up. Which is good because I didn’t want to spend the energy redoing these calculations.

The date of Thanksgiving, as observed in the United States, is that it’s the fourth Thursday of November. So it might happen anytime from the 22nd through the 28th. But because of the quirks of the Gregorian calendar, it can happen that a particular date, like the 23rd of November, is more or less likely to be a Thursday than some other day of the week.

So here’s the results of what days are most and least likely to be Thanksgiving. It turns out the 23rd, this year’s candidate, is tied for the rarest of Thanksgiving days. It’s not that rare, in comparison. It happens only two fewer times every 400 years than do Thanksgivings on the 22nd of November, the (tied) most common day.

Reading the Comics, November 18, 2017: Story Problems and Equation Blackboards Edition

It was a normal-paced week at Comic Strip Master Command. It was also one of those weeks that didn’t have anything from Comics Kingdom or Creators.Com. So I’m afraid you’ll all just have to click the links for strips you want to actually see. Sorry.

Bill Amend’s FoxTrot for the 12th has Jason and Marcus creating “mathic novels”. They, being a couple of mathematically-gifted smart people, credit mathematics knowledge with smartness. A “chiliagon” is a thousand-sided regular polygon that’s mostly of philosophical interest. A regular polygon with a thousand equal sides and a thousand equal angles looks like a circle. There’s really no way to draw one so that the human eye could see the whole figure and tell it apart from a circle. But if you can understand the idea of a regular polygon it seems like you can imagine a chilagon and see how that’s not a circle. So there’s some really easy geometry things that can’t be visualized, or at least not truly visualized, and just have to be reasoned with.

Rick Detorie’s One Big Happy for the 12th is a story-problem-subversion joke. The joke’s good enough as it is, but the supposition of the problem is that the driving does cover fifty miles in an hour. This may not be the speed the car travels at the whole time of the problem. Mister Green is maybe speeding to make up for all the time spent travelling slower.

Brandon Sheffield and Dami Lee’s Hot Comics for Cool People for the 13th uses a blackboard full of equations to represent the deep thinking being done on a silly subject.

Shannon Wheeler’s Too Much Coffee Man for the 15th also uses a blackboard full of equations to represent the deep thinking being done on a less silly subject. It’s a really good-looking blackboard full of equations, by the way. Beyond the appearance of our old friend E = mc2 there’s a lot of stuff that looks like legitimate quantum mechanics symbols there. They’re at least not obvious nonsense, as best I can tell without the ability to zoom the image in. I wonder if Wheeler didn’t find a textbook and use some problems from it for the feeling of authenticity.

Samson’s Dark Side of the Horse for the 16th is a story-problem subversion joke.

Jef Mallett’s Frazz for the 18th talks about making a bet on the World Series, which wrapped up a couple weeks ago. It raises the question: can you bet on an already known outcome? Well, sure, you can bet on anything you like, given a willing partner. But there does seem to be something fundamentally different between betting on something whose outcome isn’t in principle knowable, such as the winner of the next World Series, and betting on something that could be known but happens not to be, such as the winner of the last. We see this expressed in questions like “is it true the 13th of a month is more likely to be Friday than any other day of the week?” If you know which month and year is under discussion the chance the 13th is Friday is either 1 or 0. But we mean something more like, if we don’t know what month and year it is, what’s the chance this is a month with a Friday the 13th? Something like this is at work in this World Series bet. (The Astros won the recently completed World Series.)

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th is also featured on some underemployed philosopher’s “Reading the Comics” WordPress blog and fair enough. Utilitarianism exists in an odd triple point, somewhere on the borders of ethics, economics, and mathematics. The idea that one could quantize the good or the utility or the happiness of society, and study how actions affect it, is a strong one. It fits very well the modern mindset that holds everything can be quantified even if we don’t know how to do it well just yet. And it appeals strongly to a mathematically-minded person since it sounds like pure reason. It’s not, of course, any more than any ethical scheme can be. But it sounds like the ethics a Vulcan would come up with and that appeals to a certain kind of person. (The comic is built on one of the implications of utilitarianism that makes it seem like the idea’s gone off the rails.)

There’s some mathematics symbols on The Utilitarian’s costume. The capital U on his face is probably too obvious to need explanation. The $\sum u$ on his chest relies on some mathematical convention. For maybe a half-millennium now mathematicians have been using the capital sigma to mean “take a sum of things”. The things are whatever the expression after that symbol is. Usually, the Sigma will have something below and above which carries meaning. It says what the index is for the thing after the symbol, and what the bounds of the index are. Here, it’s not set. This is common enough, though, if this is understood from context. Or if it’s obvious. The small ‘u’ to the right suggests the utility of whatever’s thought about. (“Utility” being the name for the thing measured and maximized; it might be happiness, it might be general well-being, it might be the number of people alive.) So the symbols would suggest “take the sum of all the relevant utilities”. Which is the calculation that would be done in this case.

Reading the Comics, November 11, 2017: Pictured Comics Edition

And now the other half of last week’s comic strips. It was unusually rich in comics that come from Comics Kingdom or Creators.com, which have limited windows of access and therefore make me feel confident I should include the strips so my comments make any sense.

Rick Kirkman and Jerry Scott’s Baby Blues for the 9th mentions mathematics homework as a resolutely rage-inducing topic. It’s mathematics homework, obviously, or else it wouldn’t be mentioned around here. And even more specifically it’s Common Core mathematics homework. So it always is with attempts to teach subjects better. Especially mathematics, given how little confidence people have in their own mastery. I can’t blame parents for supposing any change to be just malice.

Bill Amend’s FoxTrot Classics for the 9th is about random numbers. As Jason says, it is hard to generate random numbers. Random numbers are a resource. Having a good source of them makes a lot of computation work. But they’re hard to make. It seems to be a contradiction to create random numbers by an algorithm. There’s reasons we accept pseudorandom numbers, or find quasirandom numbers. This strip originally ran the 16th of November, 2006.

Chris Browne’s Hagar the Horrible for the 10th is about the numerous. There’s different kinds of limits. There’s the greatest number of things we can count in an instant. There’s a limit to how long a string of digits or symbols we can remember. There’s the biggest number of things we can visualize. And “visualize” is a slippery concept. I think I have a pretty good idea what we mean when we say “a thousand” of something. I could calculate how long it took me to do something a thousand times, or to write a thousand of something. I know that it was at about a thousand words that, last A To Z sequence, I got to feeling I should wrap up any particular essay. But did I see any particular difference between word 999 and word 1,000? No; what I really knew was “about enough paragraphs” and maybe “fills just over two screens in my text editor”. So do I know what a thousand is? Anyway, we all have our limits, acknowledge them or not.

Henry Scarpelli and Craig Boldman’s Archie rerun for the 17th is about Moose’s struggle with mathematics. Just writing “more or less” doesn’t fix an erroneous answer, true. But error margins, and estimates of where an answer should be, can be good mathematics. (Part of the Common Core that many parents struggle with is making the estimate of an answer the first step, and a refined answer later. Based on what I see crossing social media, this really offends former engineering majors who miss the value in having an expected approximate answer.) It’s part of how we define limits, and derivatives, and integrals, and all of calculus. But it’s in a more precise way than Moose tries to do.

Ted Shearer’s Quincy for the 18th of September, 1978 is a story-problem joke. Some of these aren’t complicated strips.

Reading the Comics, November 8, 2017: Uses Of Mathematics Edition

Was there an uptick in mathematics-themed comic strips in the syndicated comics this past week? It depends how tight a definition of “theme” you use. I have enough to write about that I’m splitting the week’s load. And I’ve got a follow-up to that Wronski post the other day, so I’m feeling nice and full of content right now. So here goes.

Zach Weinersmith’s Saturday Morning Breakfast Cereal posted the 5th gets my week off to an annoying start. Science and mathematics and engineering people have a tendency to be smug about their subjects. And to see aptitude or interest in their subjects as virtue, or at least intelligence. (If they see a distinction between virtue and intelligence.) To presume that an interest in the field I like is a demonstration of intelligence is a pretty nasty and arrogant move.

And yes, I also dislike the attitude that school should be about training people. Teaching should be about letting people be literate with the great thoughts people have had. Mathematics has a privileged spot here. The field, as we’ve developed it, seems to build on human aptitudes for number and space. It’s easy to find useful sides to it. Doesn’t mean it’s vocational training.

Lincoln Peirce’s Big Nate on the 6th discovered mathematics puzzles. And this gave him the desire to create a new mathematical puzzle that he would use to get rich. Good luck with that. Coming up with interesting enough recreational mathematics puzzles is hard. Presenting it in a way that people will buy is another, possibly greater, challenge. It takes luck and timing and presentation, just as getting a hit song does. Sudoku, for example, spent five years in the Dell Magazine puzzle books before getting a foothold in Japanese newspapers. And then twenty years there before being noticed in the English-speaking puzzle world. Big Nate’s teacher tries to encourage him, although that doesn’t go as Mr Staples might have hoped. (The storyline continues to the 11th. Spoiler: Nate does not invent the next great recreational mathematics puzzle.)

Jef Mallett’s Frazz for the 7th start out in a mathematics class, at least. I suppose the mathematical content doesn’t matter, though. Mallett’s making a point about questions that, I confess, I’m not sure I get. I’ll leave it for wiser heads to understand.

Mike Thompson’s Grand Avenue for the 8th is a subverted word-problem joke. And I suppose a reminder about the need for word problems to parse as things people would do, or might be interested in. I can’t go along with characterizing buying twelve candy bars “gluttonous” though. Not if they’re in a pack of twelve or something like that. I may be unfair to Grand Avenue. Mind, until a few years ago I was large enough my main method of getting around was “being rolled by Oompa-Loompas”, so I could be a poor judge.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 8th does a rounding joke. It’s not much, but I’ve included appearances of this joke before and it seems unfair to skip it this time.

What Only One Person Ever Has Thought ‘Pi’ Means, And Who That Was

I’ve been reading Carl B Boyer’s The History of Calculus and its Conceptual Development. It’s been slow going, because reading about how calculus’s ideas developed is hard. The ideas underlying it are subtle to start with. And the ideas have to be discussed using vague, unclear definitions. That’s not because dumb people were making arguments. It’s because these were smart people studying ideas at the limits of what we understood. When we got clear definitions we had the fundamentals of calculus understood. (By our modern standards. The future will likely see us as accepting strange ambiguities.) And I still think Boyer whiffs the discussion of Zeno’s Paradoxes in a way that mathematics and science-types usually do. (The trouble isn’t imagining that infinite series can converge. The trouble is that things are either infinitely divisible or they’re not. Either way implies things that seem false.)

Anyway. Boyer got to a part about the early 19th century. This was when mathematicians were discovering infinities and infinitesimals are amazing tools. Also that mathematicians should maybe learn if they follow any rules. Because you can just plug symbols in to formulas and grind out what looks like they might mean and get answers. Sometimes this works great. Grind through the formulas for solving cubic polynomials as though square roots of negative numbers make sense. You get good results. Later, we worked out a coherent scheme of “complex-valued numbers” that justified it all. We can get lucky with infinities and infinitesimals, sometimes.

And this brought Boyer to an argument made by Józef Maria Hoëne-Wronski. He was a Polish mathematician whose fantastic ambition in … everything … didn’t turn out many useful results. Algebra, the Longitude Problem, building a rival to the railroad, even the Kosciuszko Uprising, none quite panned out. (And that’s not quite his name. The ‘n’ in ‘Wronski’ should have an acute mark over it. But WordPress’s HTML engine doesn’t want to imagine such a thing exists. Nor do many typesetters writing calculus or differential equations books, Boyer’s included.)

But anyone who studies differential equations knows his name, for a concept called the Wronskian. It’s a matrix determinant that anyone who studies differential equations hopes they won’t ever have to do after learning it. And, says Boyer, Wronski had this notion for an “absolute meaning of the number π”. (By “absolute” Wronski means one that not drawn from cultural factors like the weird human interset in circle perimeters and diameters. Compare it to the way we speak of “absolute temperature”, where the zero means something not particular to western European weather.)

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

Well.

I will admit I’m not fond of “real” alternate definitions of π. They seem to me mostly to signal how clever the definition-originator is. The only one I like at all defines π as the smallest positive root of the simple-harmonic-motion differential equation. (With the right starting conditions and all that.) And I’m not sure that isn’t “circumference over diameter” in a hidden form.

And yes, that definition is a mess of early-19th-century wild, untamed casualness in the use of symbols. But I admire the crazypants beauty of it. If I ever get a couple free hours I should rework it into something grammatical. And then see if, turned into something tolerable, Wronski’s idea is something even true.

Boyer allows that “perhaps” because of the strange notation and “bizarre use of the symbol ∞” Wronski didn’t make much headway on this point. I can’t fault people for looking at that and refusing to go further. But isn’t it enchanting as it is?

Reading the Comics, November 4, 2017: Slow, Small Week Edition

It was a slow week for mathematically-themed comic strips. What I have are meager examples. Small topics to discuss. The end of the week didn’t have anything even under loose standards of being on-topic. Which is fine, since I lost an afternoon of prep time to thunderstorms that rolled through town and knocked out power for hours. Who saw that coming? … If I had, I’d have written more the day before.

Mac King and Bill King’s Magic in a Minute for the 29th of October looks like a word problem. Well, it is a word problem. It looks like a problem about extrapolating a thing (price) from another thing (quantity). Well, it is an extrapolation problem. The fun is in figuring out what quantities are relevant. Now I’ve spoiled the puzzle by explaining it all so.

Olivia Walch’s Imogen Quest for the 30th doesn’t say it’s about a mathematics textbook. But it’s got to be. What other kind of textbook will have at least 28 questions in a section and only give answers to the odd-numbered problems in back? You never see that in your social studies text.

Eric the Circle for the 30th, this one by Dennill, tests how slow a week this was. I guess there’s a geometry joke in Jane Austen? I’ll trust my literate readers to tell me. My doing the world’s most casual search suggests there’s no mention of triangles in Pride and Prejudice. The previous might be the most ridiculously mathematics-nerdy thing I have written in a long while.

Tony Murphy’s It’s All About You for the 31st does some advanced-mathematics name-dropping. In so doing, it’s earned a spot taped to the door of two people in any mathematics department with more than 24 professors across the country. Or will, when they hear there was a gap unification theory joke in the comics. I’m not sure whether Murphy was thinking of anything particular in naming the subject “gap unification theory”. It sounds like a field of mathematical study. But as far as I can tell there’s just one (1) paper written that even says “gap unification theory”. It’s in partition theory. Partition theory is a rich and developed field, which seems surprising considering it’s about breaking up sets of the counting numbers into smaller sets. It seems like a time-waster game. But the game sneaks into everything, so the field turns out to be important. Gap unification, in the paper I can find, is about studying the gaps between these smaller sets.

There’s also a “band-gap unification” problem. I could accept this name being shortened to “gap unification” by people who have to say its name a lot. It’s about the physics of semiconductors, or the chemistry of semiconductors, as you like. The physics or chemistry of them is governed by the energies that electrons can have. Some of these energies are precise levels. Some of these energies are bands, continuums of possible values. When will bands converge? When will they not? Ask a materials science person. Going to say that’s not mathematics? Don’t go looking at the papers.

Whether partition theory or materials since it seems like a weird topic. Maybe Murphy just put together words that sounded mathematical. Maybe he has a friend in the field.

Bill Amend’s FoxTrot Classics for the 1st of November is aiming to be taped up to the high school teacher’s door. It’s easy to show how the square root of two is irrational. Takes a bit longer to show the square root of three is. Turns out all the counting numbers are either perfect squares — 1, 4, 9, 16, and so on — or else have irrational square roots. There’s no whole number with a square root of, like, something-and-three-quarters or something-and-85-117ths. You can show that, easily if tediously, for any particular whole number. What’s it look like to show for all the whole numbers that aren’t perfect squares already? (This strip originally ran the 8th of November, 2006.)

Guy Gilchrist’s Nancy for the 1st does an alphabet soup joke, so like I said, it’s been a slow week around here.

John Zakour and Scott Roberts’s Maria’s Day for the 2nd is really just mathematics being declared hated, so like I said, it’s been a slow week around here.

How October 2017 Treated My Mathematics Blog

October paid less attention to my mathematics blog than did September. I expected that. I published rather fewer pieces in October as the A To Z project had finished. And there’s some extent to which publishing anything is valuable in getting readership. How important I don’t know. I’ve never tried testing the relationship between how many readers I get and how many articles I post. I imagine the number of confounding factors would make their relationship vague. But I could run it anyway, as an example of how to do that kind of calculation.

It also makes me wonder whether republishing older essays is worthwhile. Or at least posting links to older content. I worry about boring longtime readers, although I’m not sure how many of those I even have. And it happens two of my most popular essays this month were fairly old bits of writing. I like to list the top five around here, but there was a three-way tie for fifth place. Big in October were:

That “here’s a thing I read” also seems to be a reliably popular post suggests maybe I need to do a weekly post about just other mathematics stuff I’d read.

United States 632
United Kingdom 76
India 67
Philippines 60
Germany 16
Slovenia 16
Singapore 15
Australia 12
France 11
Austria 10
Romania 7
Spain 7
Malaysia 6
Brazil 5
Kuwait 5
Netherlands 5
Turkey 5
Belarus 4
Hong Kong SAR China 4
Italy 4
South Africa 4
European Union 3
Poland 3
Slovakia 3
South Korea 3
Argentina 2
Denmark 2
Indonesia 2
Iraq 2
Ireland 2
Mexico 2
Norway 2
St. Kitts and Nevis 2
Sweden 2
Thailand 2
Ukraine 2
United Arab Emirates 2
Albania 1
Belgium 1 (*)
Bulgaria 1 (*)
China 1
Hungary 1
Japan 1
Latvia 1
Macedonia 1
New Zealand 1 (*)
Russia 1
Switzerland 1
Taiwan 1

I make that out to be 51 countries sending me readers at all, down from September’s 65. There were 13 single-reader countries, down from September’s 20. Belgium, Bulgaria, and New Zealand were single-reader countries for two months in a row, and no country’s on a three-month single-reader streak. “European Union” is back after a month’s absence. I’m still surprised by the number of readers from the Philippines I’ve drawn two months in a row now.

All together there were 1,069 page views from 614 unique visitors in October. That’s down from 1,232 page views and 672 unique visitors in September, and an up-and-down split from the 1,030 page views from 680 unique visitors in August. In August there were 21 posts here, in September 20, and in October 13. I kind of get the feeling people like me, but only a certain amount of me, and then they drift off.

The number of ‘likes’ went back to cratering, down to 64 over the month of October. There’d been 98 in September and 147 in August. The number of comments fell too, to a meager 12 from September’s 42 and August’s 46. The A To Z format definitely looks more inviting and welcoming to commenters, I have to conclude.

October finished out with my page here having collected 54,336 total page views from some 25,288 admitted unique visitors. I believe there were a few more visitors but some of them were copying.

Insights says that the most popular day for page views was Monday, which drew 18 percent of page views, down just a bit from September’s 20 percent. In a major upset 6 pm was not the most popular hour for readers, though. 7 pm was, when 8 percent of page views came in. I’m not sure how that happened; 6 pm is when I set most stuff to post and readers seem to follow. Maybe it’s a Daylight Saving Time issue. Oh, come to think of it, this is one of the few weeks that Greenwich Time and Eastern Time aren’t in Daylight-Saving/Summer-Time synch, isn’t it? I started out with this as a joke but perhaps that’s really going on. (No, I guess not. 12:00 am is still my most popular hour on my humor blog.) Anyway, I’m figuring to skip future mentions of what Insights tells me about popular days or hours. I can’t figure how they’re indicating anything more than “I’m about equally popular-ish any hour of any day of the week”.

WordPress says I’m starting November with 709 WordPress.com followers, which is down from September’s 717. Well, I’m sure all 709 of them are live, active accounts from people who’ve used them more recently than three years ago when they posted twice. If you’d like to follow my mathematical chats here you can add it to your reader. Go to the upper right corner of this page and click the ‘Follow NebusResearch’ button. If you’d rather get things by e-mail, there should be a ‘Follow Blog Via E-Mail’ button there too. And if that’s all fine enough but you’d like to see me limited to about 22 words at a time, try out @Nebusj on Twitter. Thanks.

This attractive little tweet came across my feed yesterday:

This function — I guess it’s the “popcorn” function — is a challenge to our ideas about what a “continuous” function is. I’ve mentioned “continuous” functions before and said something like they’re functions you could draw without lifting your pen from the paper. That’s the colloquial, and the intuitive, idea of what they mean. And that’s all right for ordinary uses.

But the best definition mathematicians have thought of for a “continuous function” has some quirks. And here’s one of them. Define a function named ‘f’. Its domain is the real numbers. Its range is the real numbers. And the rule matching things in the domain to things in the range is, as pictured:

• If ‘x’ is zero then $f(x) = 1$
• If ‘x’ is an irrational number then $f(x) = 0$
• If ‘x’ is a rational number, then it’s equal in lowest terms to the whole number ‘p’ divided by the positive whole number ‘q’. And for this ‘x’, then $f(x) = \frac{1}{q}$

And as the tweet from Fermat’s Library says, this is a function that’s continuous on all the irrational numbers. It’s not continuous on any rational numbers. This seems like a prank. But it’s a common approach to finding intuition-testing ideas about continuity. Setting different rules for rational and irrational numbers works well for making these strange functions. And thinking of rational numbers as their representation in lowest terms is also common. (Writing it as ‘p divided by q’ suggests that ‘p’ and ‘q’ are going to be prime, but, no! Think of $\frac{3}{8}$ or of $\frac{4}{9}$.) If you stare at the plot you can maybe convince yourself that “continuous on the irrational numbers” makes sense here. That heavy line of dots at the bottom looks like it’s approaching a continuous blur, at least.

It can get weirder. It’s possible to create a function that’s continuous at only a single point of all the real numbers. This is why Real Analysis is such a good subject to crash hard against. But we accept weird conclusions like this because the alternative is to give up as “continuous” functions that we just know have to be continuous. Mathematical definitions are things we make for our use.

Reading the Comics, October 2017: Mathematics Anxiety Edition

Comic Strip Master Command hasn’t had many comics exactly on mathematical points the past week. I’ll make do. There are some that are close enough for me, since I like the comics already. And enough of them circle around people being nervous about doing mathematics that I have a title for this edition.

Tony Cochrane’s Agnes for the 24th talks about math anxiety. It’s not a comic strip that will do anything to resolve anyone’s mathematics anxiety. But it’s funny about its business. Agnes usually is; it’s one of the less-appreciated deeply-bizarre comics out there.

John Atkinson’s Wrong Hands for the 24th might be the anthropomorphic numerals joke for this week. Or it might be the anthropomorphic letters joke. Or something else entirely.

Charles Schulz’s Peanuts for the 24th reruns the comic from the 2nd of November, 1970. It has Sally discovering that multiplication is much easier than she imagined. As it is, she’s not in good shape. But if you accept ‘tooty-two’ as another name for ‘four’ and ‘threety-three’ as another name for ‘nine’, why not? And she might do all right in group theory. In that you can select a bunch of things, called ‘elements’, and describe their multiplication to fit anything you like, provided there’s consistency. There could be a four-forty-four if that seems to answer some question.

Steve Kelley and Jeff Parker’s Dustin for the 25th might be tied in to mathematics anxiety. At least it expresses how the thought of mathematics will cause some people to shut down entirely. Shame for them, but I can’t deny it’s so.

Jason Poland’s Robbie and Bobby for the 26th is an anthropomorphic geometry joke. And it’s a shape joke I don’t remember seeing, at least not under my Reading the Comics line of jokes. (Maybe I’ve just forgotten). Also, trapezoids: my most popular post of all time ever, even though it’s only got a couple months’ lead on the other perennial favorite, about how many grooves are on a record’s side.

Jerry Scott and Jim Borgman’s Zits for the 27th uses mathematics as the emblem of complicated stuff in need of study. It’s a good visual. I have to say Jeremy’s material seems unorganized to start with, though.

Something I Did Read: Literature’s Greatest Opening Lines, As Written By Mathematicians

I imagine anyone likely to read my little notes here is already familiar with Ben Orlin’s great blog, but there’s always someone who missed the cool stuff earlier. And it’s going around Mathematics Twitter again so people are still learning. So here. Math With Bad Drawings — exactly what it says on the tin — had this amusing bit, Literature’s Greatest Opening Lines, as Written By Mathematicians.

My lone regret is the best one, Anna Karenina, is really only funny if you got into advanced mathematics classes. There’s several follow-ups in the comments. I’m tempted to try writing a couple myself. Do enjoy.

This is mostly a post for myself, so that I remember the existence of something I mean to read. I have tried downloading and putting into scattered files stuff I mean to read. I’ve also tried stuffing links of stuff I mean to read into Yojimbo. Maybe putting it here will at least let someone read the things.

Anyway, this is a short essay by Joel Abraham that’s on arxiv.org. It’s Introduction to the p-adic Space. p-adics are a method of thinking about what the real numbers are. Why we need ways to think about what the real numbers are turn up when you think carefully about where our idea of them comes from.

It’s easy to see where the counting numbers like ‘1’ and ‘2’ and ‘3’ come from. They’re part of our evolutionary heritage, the part of mathematics that we know is understood also by apes and crows and raccoons. We understand some of it before we even have language.

With some thinking, and many people helping, we can go from these counting numbers to the idea of ‘0’. And even into the negative counting numbers like ‘-4’. And by thinking about multiplication, and how to reverse multiplication, we get fractions. Rational numbers. Positive and negative, given the chance. But then what are the irrational numbers? We can work out easily there have to be irrational numbers. We can name some of them. But how to give a clear definition of the whole mass of them? It should be more than just “also the other numbers”.

The p-adic numbers are one of ways to go about this. They start with thinking what we mean for two numbers to be “close to” one another. And thinking hard about how to write numbers. and this gets to interesting insights I don’t know as well as I’d like.

For this deficiency I blame Usenet. I first noticed p-adics in the voluminous and not particularly wise rantings of a crank poster to sci.math, back in the day. I forget what point, if any, he was trying to prove. But to first notice a subject as someone’s apparently idiosyncratic scheme of rewriting numbers so that everything we were already used to was useless, and in the service of some clearly nonsense goal (I think he was maybe trying to show how the number meant by 0.99999… was somehow different from the number meant by 1), is to badly hobble it. And I followed strongly mathematical-physics classes as an undergraduate and a graduate student. It’s easy to just miss problems of number representation. (This although p-adics could offer some advantages in numerical computing. They could make more numerically stable representations of irrational numbers.)

As I say, I want to fix that, and a friend linked to this arxiv post. And now that I’ve said stuff about it in public maybe it’ll coax me into going back and reading and understanding it all. We’ll see.

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

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

Hector D Cantu and Carlos Castellanos’s Baldo for the 17th has Gracie quizzed about percentages of small prices, apparently as a test of her arithmetic. Her aunt has other ideas in mind. It’s hard to dispute that this is mathematics people use in real life. The commenters on GoComics got into an argument about whether Gracie gave the right answers, though. That is, not that 20 percent of $5.95 is anything about$1.19. But did Tia Carmen want to know what 20 percent of $5.95, or did she want to know what$5.95 minus 20 percent of that price was? Should Gracie have answered $4.76 instead? It took me a bit to understand what the ambiguity was, but now that I see it, I’m glad I didn’t write a multiple-choice test with both$1.19 and $4.76 as answers. I’m not sure how to word the questions to avoid ambiguity yet still sound like something one of the hew-mons might say. Dan Thompson’s Brevity for the 19th uses the blackboard and symbols on it as how a mathematician would prove something. In this case, love. Arithmetic’s a good visual way of communicating the mathematician at work here. I don’t think a mathematician would try arguing this in arithmetic, though. I mean if we take the premise at face value. I’d expect an argument in statistics, so, a mathematician showing various measures of … feelings or something. And tests to see whether it’s plausible this cluster of readings could come out by some reason other than love. If that weren’t used, I’d expect an argument in propositional logic. And that would have long strings of symbols at work, but they wouldn’t look like arithmetic. They look more like Ancient High Martian. Just saying. Reza Farazmand’s Poorly Drawn Lines for the 20th you maybe already saw going around your social media. It’s well-designed for that. Also for grad students’ office doors. Dave Coverly’s Speed Bump for the 20th is designed with crossover appeal in mind and I wonder if whoever does Reading the Comics for English Teacher Jokes is running this same strip in their collection for the week. Darrin Bell’s Candorville for the 21st sees Lemont worry that he’s forgotten how to do long division. And, fair enough: any skill you don’t use in long enough becomes stale, whether it’s division or not. You have to keep in practice and, in time, have to decide what you want to keep in practice about. (That said, I have a minor phobia about forgetting how to prove the Contraction Mapping Theorem, as several professors in grad school stressed how it must always be possible to give a coherent proof of that, even if you’re startled awake in the middle of the night by your professor.) Me, I would begin by estimating what 4,858.8 divided by 297.492 should be. 297.492 is very near 300. And 4,858.8 is a little over 4800. And that’s suggestive because it’s obvious that 48 divided by 3 is 16. Well, it’s obvious to me. So I would expect the answer to be “a little more than 16” and, indeed, it’s about 16.3. (Don’t read the comments on GoComics. There’s some slide-rule-snobbishness, and some snark about the uselessness of the skill or the dumbness of Facebook readers, and one comment about too many people knowing how to multiply by someone who’s reading bad population-bomb science fiction of the 70s.) A Summer 2017 Mathematics A To Z Appendix: Are Colbert Numbers A Thing? This is something I didn’t have space for in the proper A To Z sequence. And it’s more a question than exposition anyway. But what the heck: I like excuses to use my nice shiny art package. I was looking for mathematics topics I might write about if I didn’t get requests for particular letters. ‘C’ came up ‘cohomology’, but what if it hadn’t? I found an interesting-looking listing at MathWorld’s dictionary. The Colbert Numbers sounded interesting. this is a collection of very long prime numbers. Each of them has at least a million decimal digits. They relate to a conjecture by Wacław Sierpiński, who’s gone months without a mention around here. The conjecture is about whole numbers that are equal to $k \cdot 2^n + 1$ for some whole numbers ‘k’ and ‘n’. Are there choices of ‘k’ for which, no matter what positive whole number ‘n’ you pick, $k \cdot 2^n + 1$ is never a prime number? (‘k’ has to meet some extra conditions.) I’m not going to explain why this is interesting because I don’t know. It’s a number theory question. They’re all strange and interesting questions in their ways. If I were writing an essay about Colbert Numbers I’d have figured that out. Thing is we believe we know what the smallest possible ‘k’ is. We think that the smallest possible Sierpiński number is 78,557. We don’t have this quite proved, though. There are some numbers that might be prime numbers of the form $k \cdot 2^n + 1$ for some ‘k’ and some ‘n’. There was a set of seventeen possible candidates of numbers smaller than 78,557 that might be Sierpiński numbers. If those candidates could be ruled out then we’d have proved 78,557 was it. That’s easy to imagine. Find for each of them a number ‘n’ so that the candidate times 2n plus one was a prime number. But finding big prime numbers is hard. This turned into a distributed-computing search. This would evaluate these huge numbers and find whether they were prime numbers. (The project, “Seventeen Or Bust”, was destroyed by computer failure in 2016. Attempts to verify the work done, and continue it, are underway.) There are six possible candidates left. MathWorld says that the seventeen cases that had to be checked were named Colbert Numbers. This was in honor of Stephen T Colbert, the screamingly brilliant character host of The Colbert Report. (Ask me sometime about the Watership Down anecdote.) It’s a plausible enough claim. Part of Stephen T Colbert’s persona was demanding things be named for him. And he’d take appropriate delight in having minor but interesting things named for him. Treadmills on the space station. Minor-league hockey team mascots. A class of numbers for proving a 60-year-old mathematical conjecture is exactly the sort of thing that would get his name and attention. But here’s my problem. Who named them Colbert Numbers? MathWorld doesn’t say. Wikipedia doesn’t say. The best I can find with search engines doesn’t say. When were they named Colbert Numbers? Again, no answers. Poking around fan sites for The Colbert Report — where you’d expect the naming of stuff in his honor to be mentioned — doesn’t turn up anything. Does anyone call them Colbert Numbers? I mean outside people who’ve skimmed MathWorld’s glossary for topics with intersting names? I don’t mean to sound overly skeptical here. But, like, I know there’s a class of science fiction fans who like to explain how telerobotics engineers name their hardware “waldoes”. This is in honor of a character in a Robert Heinlein story I never read either. I’d accepted that without much interest until Google’s US Patent search became a thing. One afternoon I noticed that if telerobotics engineers do call their hardware “waldoes” they never use the term in patent applications. Is it possible that someone might have slipped a joke in to Wikipedia or something and had it taken seriously? Certainly. What amounts to a Wikipedia prank briefly gave the coati — an obscure-to-the-United-States animal that I like — the nickname of “Brazilian aardvark”. There are surely other instances of Wikipedia-generated pranks becoming “real” things. So I would like to know. Who named Colbert Numbers that, and when, and were they — as seems obvious, but you never know — named for Stephen T Colbert? Or is this an example of Wikiality, the sense that reality can be whatever enough people decide to believe is true, as described by … well, Stephen T Colbert? Reading the Comics, October 14, 2017: Physics Equations Edition So that busy Saturday I promised for the mathematically-themed comic strips? Here it is, along with a Friday that reached the lowest non-zero levels of activity. Stephan Pastis’s Pearls Before Swine for the 13th is one of those equations-of-everything jokes. Naturally it features a panel full of symbols that, to my eye, don’t parse. There are what look like syntax errors, for example, with the one that anyone could see the { mark that isn’t balanced by a }. But when someone works rough they will, often, write stuff that doesn’t quite parse. Think of it as an artist’s rough sketch of a complicated scene: the lines and anatomy may be gibberish, but if the major lines of the composition are right then all is well. Most attempts to write an equation for everything are really about writing a description of the fundamental forces of nature. We trust that it’s possible to go from a description of how gravity and electromagnetism and the nuclear forces go to, ultimately, a description of why chemistry should work and why ecologies should form and there should be societies. There are, as you might imagine, a number of assumed steps along the way. I would accept the idea that we’ll have a unification of the fundamental forces of physics this century. I’m not sure I would believe having all the steps between the fundamental forces and, say, how nerve cells develop worked out in that time. Mark Anderson’s Andertoons makes it overdue appearance for the week on the 14th, with a chalkboard word-problem joke. Amusing enough. And estimating an answer, getting it wrong, and refining it is good mathematics. It’s not just numerical mathematics that will look for an approximate solution and then refine it. As a first approximation, 15 minus 7 isn’t far off 10. And for mental arithmetic approximating 15 minus 7 as 10 is quite justifiable. It could be made more precise if a more exact answer were needed. Maria Scrivan’s Half Full for the 14th I’m going to call the anthropomorphic geometry joke for the week. If it’s not then it’s just wordplay and I’d have no business including it here. Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips for the 14th tosses in the formula describing how strong the force of gravity between two objects is. In Newtonian gravity, which is why it’s the Newton Police. It’s close enough for most purposes. I’m not sure how this supports the cause of world peace. Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 14th names Riemann’s Quaternary Conjecture. I was taken in by the panel, trying to work out what the proposed conjecture could even mean. The reason it works is that Bernhard Riemann wrote like 150,000 major works in every field of mathematics, and about 149,000 of them are big, important foundational works. The most important Riemann conjecture would be the one about zeroes of the Riemann Zeta function. This is typically called the Riemann Hypothesis. But someone could probably write a book just listing the stuff named for Riemann, and that’s got to include a bunch of very specific conjectures. Reading the Comics, October 12, 2017: Busy Saturday Soon Edition The week was looking ready to be one where I have my five paragraphs about how something shows off a word problem and that’s it. And then Comic Strip Master Command turned up the flow of comics for Saturday. So, here’s my five paragraphs about something being word problems and we’ll pick up the other half of them soon. Bill Whitehead’s Free Range for the 10th is an Albert Einstein joke. That’s usually been enough. That it mentions curved space, the exotic geometries that make general relativity so interesting, gives it a little more grounding as a mathematical comic. It’s a bit curious, surely, that curved space strikes people as so absurd. Nobody serious argues whether we live on a curved space, though, not when we see globes and think about shapes that cover a big part of the surface of the Earth. But there is something different about thinking of three-dimensional space as curved; it’s hard to imagine curved around what. Brian Basset’s Red and Rover started some word problems on the 11th, this time with trains travelling in separate directions. The word problem seemed peculiar, since the trains wouldn’t be 246 miles apart at any whole number of hours. But they will be at a reasonable fraction more than a whole number of hours, so I guess Red has gotten to division with fractions. Red and Rover are back at it the 12th with basically the same problem. This time it’s with airplanes. Also this time it’s a much worse problem. While you can do the problem still, the numbers are uglier. It’ll be just enough over two hours and ten minutes that I wonder if the numbers got rewritten away from some nicer set. For example, if the planes had been flying at 360 and 540 miles per hour, and the question was when they would be 2,100 miles apart, then you’d have a nice two-and-a-third hours. Patrick Roberts’s Todd the Dinosaur for the 12th is another in the line of jokes about fraction-teaching going wrong by picking a bad example. John Zakour and Scott Roberts’s Maria’s Day for the 12th uses mathematics as the iconic worst-possible-case for a pop quiz. I suppose spelling might have done too. The Summer 2017 Mathematics A To Z: What I Learned I’ve in the past done essays about what I’ve taken away from an A to Z project. Please indulge me with this. The big thing I learned from the Summer 2017 A To Z, besides that it would have been a little better started two weeks earlier? (I couldn’t have started it two weeks earlier. July was a frightfully busy month. As it was I was writing much too close to deadline. Starting sooner would have been impossible.) Category theory, mostly. Many of the topics requested had some category theory component. Next would be tensors and tensor-related subjects. This is exciting and dangerous. Neither’s a field I know well. Both are fields I want to know better. It’s a truism that to really learn an advanced subject you have to teach a course in it. That’s how I picked up what I know about signal processing and about numerical quantum mechanics. Still, it’s perilous, especially when I would realize the subject asked-for wasn’t what I faintly remembered had been asked for, and that I’d been composing an essay for in my head for a week already. Also, scheduling. The past A To Z sequences were relatively low-stress things for me. I could get as many as six essays ahead of what I needed to post. That’s so comfortable a place to be. This time around, I was working much closer to deadline, with some pieces needing major rewriting as few as fifteen hours before my posting hour. More needed minor editing the day of posting. There’s several causes for this. But the biggest is that I wrote much longer this time. Past A To Z sequences could have at least a couple essays that were a few paragraphs. This time around I don’t think any piece came in at under a thousand words, and the default was getting to be around 1500 words. I don’t think I broke 2,000 words, but I came close. That’s fine, because the essays came out great. This has been the A To Z sequence I’m proudest of, so far. They’re the ones that make me think my father’s ever-supportive assurance that I could put these into a book that people would give me actual money for can be right. Still, the combination of writing about stuff I had to research more first and writing longer pieces made the workload more than I’d figured on. When I get to doing this again — and I will, when the exhaustion’s faded enough from memory — I will need more lead time between asking for topics and starting to write. And will need to freeze topics farther in advance than I did this time. I still suspect my father’s too supportive to say I could get money for this. But it’s a less unrealistic thought than I had figured before. Also learned: hire an artist! I got a better-banner-than-I-paid-for from Thomas K Dye for this series. His work added a snappy bit of visual appeal to my sentence heaps. I’d also gotten from him a banner for the Why Stuff Can Orbit sequence, which I mean to resume now that I have some more writing time. But the banners give a needed bit of unity to my writing, and the automatically-generated Twitter announcements of these posts, and that’s helped the look of the place. Something like nine-tenths of the people I know online are visual artists of one kind or another. (The rest are writers, my siblings, and my mother.) I should be making reasons to commission them. For example, if I want to describe something too complicated to do in words alone I should turn it over to them. Remember, I don’t do the few-pictures thing because I’m a good writer. It’s because I’m too lazy to make an illustration myself. A bit of money can be as good as effort. Speaking of effort, between the A To Z essays and Reading the Comics posts, and a couple miscellaneous other pieces, I wrote five to six thousand words per week for two months. That’s probably not sustainable indefinitely, but a slightly lower pace? And for a specific big project? It’s good to know that’s something I can do, albeit possibly by putting this blog on hold. Learned to my personal everlasting humiliation: I spelled “Klein Bottle” wrong. Fortunately, I only spelled it “Klien” in the title of the essay, so it sits there in my tweet publicizing the post and in the full-length URL to the post, forever. I’ll recover, I hope. Reading the Comics, October 7, 2017: Rerun Comics Edition The most interesting mathematically-themed comic strips from last week were also reruns. So be it; at least I have an excuse to show a 1931-vintage comic. Also, after discovering my old theme didn’t show the category of essay I was posting, I did literally minutes of search for a new theme that did. And that showed tags. And that didn’t put a weird color behind LaTeX inline equations. So I’m using the same theme as my humor blog does, albeit with a different typeface, and we’ll hope that means I don’t post stuff to the wrong blog. As it is I start posting something to the wrong place about once every twenty times. All I want is a WordPress theme with all the good traits of the themes I look at and none of the drawbacks; why is that so hard to get? Elzie Segar’s Thimble Theatre rerun for the 5th originally ran the 25th of April, 1931. It’s just a joke about Popeye not being good at bookkeeping. In the story, Popeye’s taking the$50,000 reward from his last adventure and opened a One-Way Bank, giving people whatever money they say they need. And now you understand how the first panel of the last row has several jokes in it. The strip is partly a joke about Popeye being better with stuff he can hit than anything else, of course. I wonder if there’s an old stereotype of sailors being bad at arithmetic. I remember reading about pirate crews that, for example, not-as-canny-as-they-think sailors would demand a fortieth or a fiftieth of the prizes as their pay, instead of a mere thirtieth. But it’s so hard to tell what really happened and what’s just a story about the stupidity of people. Marginal? Maybe, but I’m a Popeye fan and this is my blog, so there.

Bill Rechin’s Crock rerun(?) from the 6th must have come before. I don’t know when. Anyway it’s a joke about mathematics being way above everybody’s head.

Norm Feuti’s Gil rerun for the 6th is a subverted word problem joke. And it’s a reminder of how hard story problems can be. You need something that has a mathematics question on point. And the question has to be framed as asking something someone would actually care to learn. Plus the story has to make sense. Much easier when you’re teaching calculus, I think.

Jason Chatfield’s Ginger Meggs for the 6th is a playing-stupid joke built in percentages. Cute enough for the time it takes to read.

Gary Wise and Lance Aldrich’s Real Life Adventures for the 6th is a parent-can’t-help-with-homework joke, done with arithmetic since it’s hard to figure another subject that would make the joke possible. I suppose a spelling assignment could be made to work. But that would be hard to write so it didn’t seem contrived.

Thaves’ Frank and Ernest for the 7th feels like it’s a riff on the old saw about Plato’s Academy. (The young royal sent home with a coin because he asked what the use of this instruction was, and since he must get something from everything, here’s his drachma.) Maybe. Or it’s just the joke that you make if you have “division” and “royals” in mind.

Mark Tatulli’s Lio for the 7th is not quite the anthropomorphic symbols joke for this past week. It’s circling that territory, though.

It was another busy week in mathematically-themed comic strips last week. Busy enough I’m comfortable rating some as too minor to include. So it’s another week where I post two of these Reading the Comics roundups, which is fine, as I’m still recuperating from the Summer 2017 A To Z project. This first half of the week includes a lot of rerun comics, and you’ll see why my choice of title makes sense.

Lincoln Pierce’s Big Nate: First Class for the 1st of October reprints the strip from the 2nd of October, 1993. It’s got a well-formed story problem that, in the time-honored tradition of this setup, is subverted. I admit I kind of miss the days when exams would have problems typed out in monospace like this.

Ashleigh Brilliant’s Pot-Shots for the 1st is a rerun from sometime in 1975. And it’s an example of the time-honored tradition of specifying how many statistics are made up. Here it comes in at 43 percent of statistics being “totally worthless” and I’m curious how the number attached to this form of joke changes over time.

The Joey Alison Sayers Comic for the 2nd uses a blackboard with mathematics — a bit of algebra and a drawing of a sphere — as the designation for genius. That’s all I have to say about this. I remember being set straight about the difference between ponies and horses and it wasn’t by my sister, who’s got a professional interest in the subject.

Mark Pett’s Lucky Cow rerun for the 2nd is a joke about cashiers trying to work out change. As one of the GoComics.com commenters mentions, the probably best way to do this is to count up from the purchase to the amount you have to give change for. That is, work out $12.43 to$12.50 is seven cents, then from $12.50 to$13.00 is fifty more cents (57 cents total), then from $13.00 to$20.00 is seven dollars ($7.57 total) and then from$20 to $50 is thirty dollars ($37.57 total).

It does make me wonder, though: what did Neil enter as the amount tendered, if it wasn’t $50? Maybe he hit “exact change” or whatever the equivalent was. It’s been a long, long time since I worked a cash register job and while I would occasionally type in the wrong amount of money, the kinds of errors I would make would be easy to correct for. (Entering$30 instead of \$20 for the tendered amount, that sort of thing.) But the cash register works however Mark Pett decides it works, so who am I to argue?

Keith Robinson’s Making It rerun for the 2nd includes a fair bit of talk about ratios and percentages, and how to inflate percentages. Also about the underpaying of employees by employers.

Mark Anderson’s Andertoons for the 3rd continues the streak of being Mark Anderson Andertoons for this sort of thing. It has the traditional form of the student explaining why the teacher’s wrong to say the answer was wrong.

Brian Fies’s The Last Mechanical Monster for the 4th includes a bit of legitimate physics in the mad scientist’s captioning. Ballistic arcs are about a thing given an initial speed in a particular direction, moving under constant gravity, without any of the complicating problems of the world involved. No air resistance, no curvature of the Earth, level surfaces to land on, and so on. So, if you start from a given height (‘y0‘) and a given speed (‘v’) at a given angle (‘θ’) when the gravity is a given strength (‘g’), how far will you travel? That’s ‘d’. How long will you travel? That’s ‘t’, as worked out here.

(I should maybe explain the story. The mad scientist here is the one from the first, Fleischer Studios, Superman cartoon. In it the mad scientist sends mechanical monsters out to loot the city’s treasures and whatnot. As the cartoon has passed into the public domain, Brian Fies is telling a story of that mad scientist, finally out of jail, salvaging the one remaining usable robot. Here, training the robot to push aside bank tellers has gone awry. Also, the ground in his lair is not level.)

Tom Toles’s Randolph Itch, 2 am rerun for the 4th uses the time-honored tradition of Albert Einstein needing a bit of help for his work.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 4th uses the time-honored tradition of little bits of physics equations as designation of many deep thoughts. And then it gets into a bit more pure mathematics along the way. It also reflects the time-honored tradition of people who like mathematics and physics supposing that those are the deepest and most important kinds of thoughts to have. But I suppose we all figure the things we do best are the things it’s important to do best. It’s traditional.

And by the way, if you’d like more of these Reading the Comics posts, I put them all in the category ‘Comic Strips’ and I just now learned the theme I use doesn’t show categories for some reason? This is unsettling and unpleasant. Hm.

Reading the Comics, September 29, 2017: Anthropomorphic Mathematics Edition

The rest of last week had more mathematically-themed comic strips than Sunday alone did. As sometimes happens, I noticed an objectively unimportant detail in one of the comics and got to thinking about it. Whether I could solve the equation as posted, or whether at least part of it made sense as a mathematics problem. Well, you’ll see.

Patrick McDonnell’s Mutts for the 25th of September I include because it’s cute and I like when I can feature some comic in these roundups. Maybe there’s some discussion that could be had about what “equals” means in ordinary English versus what it means in mathematics. But I admit that’s a stretch.

Olivia Walch’s Imogen Quest for the 25th uses, and describes, the mathematics of a famous probability problem. This is the surprising result of how few people you need to have a 50 percent chance that some pair of people have a birthday in common. It then goes over to some other probability problems. The examples are silly. But the reasoning is sound. And the approach is useful. To find the chance of something happens it’s often easiest to work out the chance it doesn’t. Which is as good as knowing the chance it does, since a thing can either happen or not happen. At least in probability problems, which define “thing” and “happen” so there’s not ambiguity about whether it happened or not.

Piers Baker’s Ollie and Quentin rerun for the 26th I’m pretty sure I’ve written about before, although back before I included pictures of the Comics Kingdom strips. (The strip moved from Comics Kingdom over to GoComics, which I haven’t caught removing old comics from their pages.) Anyway, it plays on a core piece of probability. It sets out the world as things, “events”, that can have one of multiple outcomes, and which must have one of those outcomes. Coin tossing is taken to mean, by default, an event that has exactly two possible outcomes, each equally likely. And that is near enough true for real-world coin tossing. But there is a little gap between “near enough” and “true”.

Rick Stromoski’s Soup To Nutz for the 27th is your standard sort of Dumb Royboy joke, in this case about him not knowing what percentages are. You could do the same joke about fractions, including with the same breakdown of what part of the mathematics geek population ruins it for the remainder.

Nate Fakes’s Break of Day for the 28th is not quite the anthropomorphic-numerals joke for the week. Anthropomorphic mathematics problems, anyway. The intriguing thing to me is that the difficult, calculus, problem looks almost legitimate to me. On the right-hand-side of the first two lines, for example, the calculation goes from

$\int -8 e^{-\frac{ln 3}{14} t}$

to
$-8 -\frac{14}{ln 3} e^{-\frac{ln 3}{14} t}$

This is a little sloppy. The first line ought to end in a ‘dt’, and the second ought to have a constant of integration. If you don’t know what these calculus things are let me explain: they’re calculus things. You need to include them to express the work correctly. But if you’re just doing a quick check of something, the mathematical equivalent of a very rough preliminary sketch, it’s common enough to leave that out.

It doesn’t quite parse or mean anything precisely as it is. But it looks like the sort of thing that some context would make meaningful. That there’s repeated appearances of $- \frac{ln 3}{14}$, or $- \frac{14}{ln 3}$, particularly makes me wonder if Frakes used a problem he (or a friend) was doing for some reason.

Mark Anderson’s Andertoons for the 29th is a welcome reassurance that something like normality still exists. Something something student blackboard story problem something.

Anthony Blades’s Bewley rerun for the 29th depicts a parent once again too eager to help with arithmetic homework.

Maria Scrivan’s Half Full for the 29th gives me a proper anthropomorphic numerals panel for the week, and none too soon.