There is this thing called the abc Conjecture. It’s a big question in number theory, which is the part of mathematics where we learn we don’t understand anything about prime numbers. Nearly a decade ago Shinichi Mochizuki announced a proof. It’s been controversial. Most importantly, it’s not been well-understood.
It’s finally getting published in a proper journal. A lot of mathematics work is passed around as PDFs, usually on arXiv.org, these days. It’s good for sharing fresh thoughts. But journal publication usually means that the paper has been reviewed, critically, and approved by people who could tell whether the reasoning is sound. Mochizuki’s paper is somewhere around 500 to 600 pages (I’ve seen different figures), and by every report hard to understand even for number theory proofs. A proof is, more than mathematicians like to admit, really an argument that convinces other mathematicians that, if we wanted to spend the time, we could find a completely rigorous proof. With very long proofs, and very complicated proofs, the standard of being convincing gets tougher.
I know; I’m more than a week behind the original publication of these strips. The Playful Math Education Blog Carnival took a lot of what attention I have these days. I’ll get caught up again soon enough. Comic Strip Master Command tried to help me, by having the close of a week ago being pretty small mathematics mentions, too. For example:
Craig Boldman and Henry Scarpelli’s Archie for the 27th has Moose struggling in mathematics this term. This is an interesting casual mention; the joke, of Moose using three words to describe a thing he said he could in two, would not fit sharply for anything but mathematics. Or, possibly, a measuring class, but there’s no high school even in fiction that has a class in measuring.
Bud Blake’s Vintage Tiger for the 27th has Tiger and Hugo struggling to find adjective forms for numbers. We can giggle at Hugo struggling for “quadruple” and going for something that makes more sense. We all top out somewhere, though, probably around quintuple or sextuple. I have never known anyone who claimed to know what the word would be for anything past decuple, and even staring at the dictionary page for “decuple” I don’t feel confident in it.
Hilary Price’s Rhymes With Orange for the 28th uses a blackboard full of calculations as shorthand for real insight into science. From context they’re likely working on some physics problem and it’s quite hard to do that without mathematics, must agree.
John Deering’s Strange Brew for the 28th name-drops slide rules, which, yeah, have mostly historical or symbolic importance these days. There might be some niche where they’re particularly useful (besides teaching logarithms), but I don’t know of it.
I haven’t forgotten about the comic strips. It happens that last week’s were mostly quite casual mentions, strips that don’t open themselves up to deep discussions. I write this before I see what I actually have to write about the strips. But here’s the first half of the past week’s. I’ll catch up on things soon.
Bill Amend’s FoxTrot for the 22nd, a new strip, has Jason and Marcus using arithmetic problems to signal pitches. At heart, the signals between a pitcher and catcher are just an index. They’re numbers because that’s an easy thing to signal given that one only has fingers and that they should be visually concealed. I would worry, in a pattern as complicated as these two would work out, about error correction. If one signal is mis-read — as will happen — how do they recognize it, and how do they fix it? This may seem like a lot of work to put to a trivial problem, but to conceal a message is important, whatever the message is.
James Beutel’s Banana Triangle for the 23rd has a character trying to convince himself os his intelligence. And doing so by muttering mathematics terms, mostly geometry. It’s a common shorthand to represent deep thinking.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 25th is a joke about orders of magnitude. The order of magnitude is, roughly, how big the number is. Often the first step of a physics problem is to try to get a calculation that’s of the right order of magnitude. Or at least close to the order of magnitude. This may seem pretty lax. If we want to find out something with value, say, 231, it seems weird to claim victory that our model says “it will be a three-digit number”. But getting the size of the number right is a first step. For many problems, particularly in cosmology or astrophysics, we’re intersted in things whose functioning is obscure. And relies on quantities we can measure very poorly. This is why we can see getting the order magnitude about right as an accomplishment.
Greetings, friends, and thank you for visiting the 136th installment of Denise Gaskins’s Playful Math Education Blog Carnival. I apologize ahead of time that this will not be the merriest of carnivals. It has not been the merriest of months, even with it hosting Pi Day at the center.
In consideration of that, let me lead with Art in the Time of Transformation by Paula Beardell Krieg. This is from the blog Playful Bookbinding and Paper Works. The post particularly reflects on the importance of creating a thing in a time of trouble. There is great beauty to find, and make, in symmetries, and rotations, and translations. Simple polygons patterned by simple rules can be accessible to anyone. Studying just how these symmetries and other traits work leads to important mathematics. Thus how Kreig’s page has recent posts with names like “Frieze Symmetry Group F7” but also to how symmetry is for five-year-olds. I am grateful to Goldenoj for the reference.
That link was brought to my attention by Iva Sallay, another longtime friend of my little writings here. She writes fun pieces about every counting number, along with recreational puzzles. And asked to share 1458 Tangrams Can Be A Pot of Gold, as an example of what fascinating things can be found in any number. This includes a tangram. Tangrams we see in recreational-mathematics puzzles based on ways that you can recombine shapes. It’s always exciting to be able to shift between arithmetic and shapes. And that leads to a video and related thread again pointed to me by goldenoj …
This video, by Mathologer on YouTube, explains a bit of number theory. Number theory is the field of asking easy questions about whole numbers, and then learning that the answers are almost impossible to find. I exaggerate, but it does often involve questions that just suppose you understand what a prime number should be. And then, as the title asks, take centuries to prove.
Fermat’s Two-Squares Theorem, discussed here, is not the famous one about . Pierre de Fermat had a lot of theorems, some of which he proved. This one is about prime numbers, though, and particularly prime numbers that are one more than a multiple of four. This means it’s sometimes called Fermat’s 4k+1 Theorem, which is the name I remember learning it under. (k is so often a shorthand for “some counting number” that people don’t bother specifying it, the way we don’t bother to say “x is an unknown number”.) The normal proofs of this we do in the courses that convince people they’re actually not mathematics majors.
What the video offers is a wonderful alternate approach. It turns key parts of the proof into geometry, into visual statements. Into sliding tiles around and noticing patterns. It’s also a great demonstration of one standard problem-solving tool. This is to look at a related, different problem that’s easier to say things about. This leads to what seems like a long path from the original question. But it’s worth it because the path involves thinking out things like “is the count of this thing odd or even”? And that’s mathematics that you can do as soon as you can understand the question.
I again thank Iva Sallay for that link, as well as this essay. Dan Meyer’s But Artichokes Aren’t Pinecones: What Do You Do With Wrong Answers? looks at the problem of students giving wrong answers. There is no avoiding giving wrong answers. A parent’s or teacher’s response to wrong answers will vary, though, and Meyer asks why that is. Meyer has some hypotheses. His example notes that he doesn’t mind a child misidentifying an artichoke as a pinecone. Not in the same way identifying the sum of 1 and 9 as 30 would. What is different about those mistakes?
Jessannwa’s Soft Start In The Intermediate Classroom looks to the teaching of older students. No muffins and cookies here. That the students might be more advanced doesn’t change the need to think of what they have energy for, and interest in. She discusses a class setup that’s meant to provide structure in ways that don’t feel so authority-driven. And ways to turn practicing mathematics problems into optimizing game play. I will admit this is a translation of the problem which would have worked well for me. But I also know that not everybody sees a game as, in part, something to play at maximum efficiency. It depends on the game, though. They’re on Twitter as @jesannwa.
These are thoughts about how anyone can start learning mathematics. What does it look like to have learned a great deal, though, to the point of becoming renowned for it? Life Through A Mathematician’s Eyes posted Australian Mathematicians in late January. It’s a dozen biographical sketches of Australian mathematicians. It also matches each to charities or other public-works organizations. They were trying to help the continent through the troubles it had even before the pandemic struck. They’re in no less need for all that we’re exhausted. The page’s author is on Twitter as @lthmath.
I have since the start of this post avoided mentioning the big mathematical holiday of March. Pi Day had the bad luck to fall on a weekend this year, and then was further hit by the Covid-19 pandemic forcing the shutdown of many schools. Iva Sallay again helped me by noting YummyMath’s activities page It’s Time To Gear Up For Pi Day. This hosts several worksheets, about the history of π and ways to calculate it, and several formulas for π. This even gets into interesting techniques like how to use continued fractions in finding a numerical value.
Rolands Rag Bag shared A Pi-Ku for Pi-Day featuring a poem written in a form I wasn’t aware anyone did. The “Pi-Ku” as named here has 3 syllables for the first time, 1 syllable in the second line, 4 syllables in the third line, 1 syllable the next line, 5 syllables after that … you see the pattern. (One of Avery’s older poems also keeps this form.) The form could, I suppose, go on to as many lines as one likes. Or at least to the 40th line, when we would need a line of zero syllables. Probably one would make up a rule to cover that.
I hope to publish the March 2020 Playful Math Education Blog Carnival tomorrow. If you’ve recently seen any web site that shares and explains some aspect of mathematics or mathematics education that interested or delighted you, please, share it with me, so I can share it with more people. If you do, please, let me know of your own projects, besides that, so I can mention that to this month’s audience. Thank you.
There were a handful of other comic strips last week. If they have a common theme (and I’ll try to drag one out) it’s that they circle around pragmatism. Not just using mathematics in the real world but the fussy stuff of what you can calculate and what you can use a calculation for.
And, again, I am hosting the Playful Math Education Blog Carnival this month. If you’ve run across any online tool that teaches mathematics, or highlights some delightful feature of mathematics? Please, let me know about it here, and let me know what of your own projects I should feature with it. The goal is to share things about mathematics that helped you understand more of it. Even if you think it’s a slight thing (“who cares if you can tell whether a number’s divisible by 11 by counting the digits right?”) don’t worry. Slight things count. Speaking of which …
Jef Mallett’s Frazz for the 20th has a kid ask about one of those add-the-digits divisibility tests. What happens if the number is too big to add up all the digits? In some sense, the question is meaningless. We can imagine finding the sum of digits no matter how many digits there are. At least if there are finitely many digits.
But there is a serious mathematical question here. We accept the existence of numbers so big no human being could ever know their precise value. At least, we accept they exist in the same way that “4” exists. If a computation can’t actually be finished, then, does it actually mean anything? And if we can’t figure a way to shorten the calculation, the way we can usually turn the infinitely-long sum of a series into a neat little formula?
This gets into some cutting-edge mathematics. For calculations, some. But also, importantly, for proofs. A proof is, really, a convincing argument that something is true. The ideal of this is a completely filled-out string of logical deductions. These will take a long while. But, as long as it takes finitely many steps to complete, we normally accept the proof as done. We can imagine proofs that take more steps to complete than could possibly be thought out, or checked, or confirmed. We, living in the days after Gödel, are aware of the idea that there are statements which are true but unprovable. This is not that. Gödel’s Incompleteness Theorems tell us about statements that a deductive system can’t address. This is different. This is things that could be proven true (or false), if only the universe were more vast than it is.
There are logicians who work on the problem of what too-long-for-the-universe proofs can mean. Or even what infinitely long proofs can mean, if we allow those. And how they challenge our ideas of what “proof” and “knowledge” and “truth” are. I am not among these people, though, and can’t tell you what interesting results they have concluded. I just want to let you know the kid in Frazz is asking a question you can get a spot in a mathematics or philosophy department pondering. I mean so far as it’s possible to get a spot in a mathematics or philosophy department.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 20th is a less heady topic. Its speaker is doing an ethical calculation. These sorts of things are easy to spin into awful conclusions. They treat things like suffering with the same tools that we use to address the rates of fluids mixing, or of video game statistics. This often seems to trivialize suffering, which we feel like we shouldn’t do.
This kind of calculation is often done, though. It’s rather a hallmark of utilitarianism to try writing an equation for an ethical question. It blends often more into economics, where the questions can seem less cruel even if they are still about questions of life and death. But as with any model, what you build into the model directs your results. The lecturer here supposes that guilt is diminished by involving more people. (This seems rather true to human psychology, though it’s likely more that the sense of individual responsibility dissolves in a large enough group. There are many other things at work, though, all complicated and interacting in nonlinear ways.) If we supposed that the important measure was responsibility for the killing, we would get that the more people involved in killing, the worse it is, and that a larger war only gets less and less ethical. (This also seems true to human psychology.)
Jeff Corriveau’s Deflocked for the 20th sees Mamet calculating how many days of life he expects to have left. There are roughly 1,100 days in three years, so, Mamet’s figuring on about 40 years of life. These kinds of calculation are often grim to consider. But we all have long-term plans that we would like to do (retirement, and its needed savings, are an important one) and there’s no making a meaningful plan without an idea of what the goals are.
This finally closes out the last week’s comic strips. Please stop in next week as I get to some more mathematics comics and the Playful Math Education Blog Carnival. Thanks for reading.
I thought last week’s comic strips mentioning mathematics in detail were still subjects easy to describe in one or two paragraphs each. I wasn’t quite right. So here’s a half of a week, even if it is a day later than I had wanted to post.
Lincoln Peirce’s Big Nate for the 15th is a wordy bit of Nate refusing the story problem. Nate complains about a lack of motivation for the characters in it. But then what we need for a story problem isn’t the characters to do something so much as it is the student to want to solve the problem. That’s hard work. Everyone’s fascinated by some mathematical problems, but it’s hard to think of something that will compel everyone to wonder what the answer could be.
At one point Nate wonders what happens if Todd stops for gas. Here he’s just ignoring the premise of the question: Todd is given as travelling an average 55 mph until he reaches Saint Louis, and that’s that. So this question at least is answered. But he might need advice to see how it’s implied.
So this problem is doable by long division: 1825 divided by 80, and 1192 divided by 55, and see what’s larger. Can we avoid dividing by 55 if we’re doing it by hand? I think so. Here’s what I see: 1825 divided by 80 is equal to 1600 divided by 80 plus 225 divided by 80. That first is 20; that second is … eh. It’s a little less than 240 divided by 80, which is 3. So Mandy will need a little under 23 hours.
Is 23 hours enough for Todd to get to Saint Louis? Well, 23 times 55 will be 23 times 50 plus 23 times 5. 23 times 50 is 22 times 50 plus 1 times 50. 22 times 50 is 11 times 100, or 1100. So 23 times 50 is 1150. And 23 times 5 has to be 150. That’s more than 1192. So Todd gets there first. I might want to figure just how much less than 23 hours Mandy needs, to be sure of my calculation, but this is how I do it without putting 55 into an ugly number like 1192.
Mark Leiknes’s Cow and Boy repeat for the 17th sees the Boy, Billy, trying to beat the lottery. He throws at it the terms chaos theory and nonlinear dynamical systems. They’re good and probably relevant systems. A “dynamical system” is what you’d guess from the name: a collection of things whose properties keep changing. They change because of other things in the collection. When “nonlinear” crops up in mathematics it means “oh but such a pain to deal with”. It has a more precise definition, but this is its meaning. More precisely: in a linear system, a change in the initial setup makes a proportional change in the outcome. If Todd drove to Saint Louis on a path two percent longer, he’d need two percent more time to get there. A nonlinear system doesn’t guarantee that; a two percent longer drive might take ten percent longer, or one-quarter the time, or some other weirdness. Nonlinear systems are really good for giving numbers that look random. There’ll be so many little factors that make non-negligible results that they can’t be predicted in any useful time. This is good for drawing number balls for a lottery.
Chaos theory turns up a lot in dynamical systems. Dynamical systems, even nonlinear ones, often have regions that behave in predictable patterns. We may not be able to say what tomorrow’s weather will be exactly, but we can say whether it’ll be hot or freezing. But dynamical systems can have regions where no prediction is possible. Not because they don’t follow predictable rules. But because any perturbation, however small, produces changes that overwhelm the forecast. This includes the difference between any possible real-world measurement and the real quantity.
Obvious question: how is there anything to study in chaos theory, then? Is it all just people looking at complicated systems and saying, yup, we’re done here? Usually the questions turn on problems such as how probable it is we’re in a chaotic region. Or what factors influence whether the system is chaotic, and how much of it is chaotic. Even if we can’t say what will happen, we can usually say something about when we can’t say what will happen, and why. Anyway if Billy does believe the lottery is chaotic, there’s not a lot he can be doing with predicting winning numbers from it. Cow’s skepticism is fair.
Ryan North’s Dinosaur Comics for the 17th is one about people asked to summon random numbers. Utahraptor is absolutely right. People are terrible at calling out random numbers. We’re more likely to summon odd numbers than we should be. We shy away from generating strings of numbers. We’d feel weird offering, say, 1234, though that’s as good a four-digit number as 1753. And to offer 2222 would feel really weird. Part of this is that there’s not really such a thing as “a” random number; it’s sequences of numbers that are random. We just pick a number from a random sequence. And we’re terrible at producing random sequences. Here’s one study, challenging people to produce digits from 1 through 9. Are their sequences predictable? If the numbers were uniformly distributed from 1 through 9, then any prediction of the next digit in a sequence should have a one chance in nine of being right. It turns out human-generated sequences form patterns that could be forecast, on average, 27% of the time. Individual cases could get forecast 45% of the time.
There are some neat side results from that study too, particularly that they were able to pretty reliably tell the difference between two individuals by their “random” sequences. We may be bad at thinking up random numbers but the details of how we’re bad can be unique.
Justin Boyd’s Invisible Bread for the 18th> has an exhausted student making the calculation of they’ll do better enough after a good night’s sleep to accept a late penalty. This is always a difficult calculation to make, since you make it when your thinking is clouded by fatigue. But: there is no problem you have which sleep deprivation makes better. Put sleep first. Budget the rest of your day around that. Take it from one who knows and regrets a lot of nights cheated of rest. (This seems to be the first time I’ve mentioned Invisible Bread around here. Given the strip’s subject matter that’s a surprise, but only a small one.)
One of Gary Larson’s The Far Side reruns for the 19th is set in a mathematics department, and features writing a nasty note “in mathematics”. There are many mathematical jokes, some of them written as equations. A mathematician will recognize them pretty well. None have the connotation of, oh, “Kick Me” or something else that would belong as a prank sign like that. Or at least nobody’s told me about them.
Pi Day was observed with fewer, and fewer on-point, comic strips than I had expected. It’s possible that the whimsy of the day has been exhausted. Or that Comic Strip Master Command advised people that the educational purposes of the day were going to be diffused because of the accident of the calendar. And a fair number of the strips that did run in the back half of last week weren’t substantial. So here’s what did run.
And now we get to the strips that actually ran on the 14th of March.
Hector D Cantú and Carlos Castellanos’s Baldo is a slightly weird one. It’s about Gracie reflecting on how much she’s struggled with mathematics problems. There are a couple pieces meant to be funny here. One is the use of oddball numbers like 1.39 or 6.23 instead of easy-to-work-with numbers like “a dollar” or “a nickel” or such. The other is that the joke is .. something in the vein of “I thought I was wrong once, but I was mistaken”. Gracie’s calculation indicates she thinks she’s struggled with a math problem a little under 0.045 times. It’s a peculiar number. Either she’s boasting that she struggles very little with mathematics, or she’s got her calculations completely wrong and hasn’t recognized it. She’s consistently portrayed as an excellent student, though. So the “barely struggles” or maybe “only struggles a tiny bit at the start of a problem” interpretation is more likely what’s meant.
π has infinitely many decimal digits, certainly. Of course, so does 2. It’s just that 2 has boring decimal digits. Rational numbers end up repeating some set of digits. It can be a long string of digits. But it’s finitely many, and compared to an infinitely long and unpredictable string, what’s that? π we know is a transcendental number. Its decimal digits go on in a sequence that never ends and never repeats itself fully, although finite sequences within it will repeat. It’s one of the handful of numbers we find interesting for reasons other than their being transcendental. This though nearly every real number is transcendental. I think any mathematician would bet that it is a normal number, but we don’t know that it is. I’m not aware of any numbers we know to be normal and that we care about for any reason other than their normality. And this, weirdly, also despite that we know nearly every real number is normal.
Dave Whamond’s Reality Check plays on the pun between π and pie, and uses the couple of decimal digits of π that most people know as part of the joke. It’s not an anthropomorphic numerals joke, but it is circling that territory.
Michael Cavna’s Warped celebrates Albert Einstein’s birthday. This is of marginal mathematics content, but Einstein did write compose one of the few equations that an average lay person could be expected to recognize. It happens that he was born the 14th of March and that’s, in recent years, gotten merged into Pi Day observances.
Oh, yes, so. I did intend to review my readership around here last month. It’s just that things got in the way. Most of them not related to the Covid-19 pandemic; it’s much more been personal matters and my paying job and such. If someone is interested in paying me to observe that I had readers WordPress records as coming merely from the European Union, drop me a note. We can work something out. Heck, slip me ten bucks and I’ll write an essay on any mathematics topic I don’t feel wholly incompetent to discuss. Or wait around for the 2020 Mathematics A-to-Z, coming whenever I do feel up to it.
Also, do please remember that I’m hosting the Playful Math Education Blog Carnival at the end of this month. If you’ve spotted anything on the web — blog, static web site, video, podcast — that enlightened you about some field of mathematics, please let me know. And let me know of your own projects. It’ll be fun.
Now to see what my readership was like back in February, impossibly long ago as that does seem to be.
I posted 11 things in February. January had been 10. There were 1,419 page views in February. That’s just about what January was. It’s below the twelve-month running average of 2,060.3 page views. This looks dire, but it’s about the same as January’s readership. And the twelve-month average does have that anomalous October spike messing things up. If we pretend that October didn’t happen, well, that mean was something like 1460 page views.
There were 991 unique visitors in February. That’s again rather below the twelve-month running average of 1401.1 unique visitors. But again if we pretend there was no October, then the running average was something like 950 unique visitors, so things aren’t all that dire. Just that the occasional taste of popularity spoils you for ages to come.
A mere 36 things got likes here in February, below the running average of 64.1 and I’m not working out what that is with October included. Most of that readership spike didn’t convert to likes or comments anyway. Those were well-liked months but they were also ones that got something posted every single day. There were 12 comments in February, roughly in line with the 13.8 comments running average.
Per post, all these figures look a bit better. There were 129 views per posting, just over the 116.6 running average. There were 90.1 unique visitors per posting, above the running average of 78.6. There were 3.3 likes per posting, below the anemic average of 4.1. There were even 1.1 comments per posting, technically above the average of 0.9. If I could just post something four times per day that October peak would be merely an average month.
The most popular postings in February were mostly the usual suspects. Just one surprised me with its appearance:
There were 210 pages that got any views at all in February, close to the 218 of January. 108 of them got more than one view, just about the same as January’s 102. 25 pages got at least ten views. The previous couple months saw 23 and 27 posts that popular.
67 countries or country-like entities sent me any readers at all in February. That’s up from 63 in January and 60 in December. 19 of them were single-view countries, up from January’s 15 and December’s 18. Here’s the roster:
Hong Kong SAR China
United Arab Emirates
Northern Mariana Islands
Cambodia and South Korea were single-view countries in January also. American Samoa, Saudi Arabia, and Slovakia have been single-view countries for three months.
In February I posted 9,699 words by WordPress’s counter. That’s 881.7 words per posting. For the year my average post, as of the start of the month, was 755.1 words per post. Some months are talky. I had started the month with 100,432 page views, just missing out on being number 100,000 myself. And these came from a logged 54,920 unique visitors. And I had posted a total of 1,424 things from the dawn of time to the 1st of March, which by some strange fluke was itself fifty thousand years ago.
Well, I hope to keep posting and to review March a little closer to the 1st of April, which looks to be about fifty thousand years in the future. To follow along with me, add the feed https://nebusresearch.wordpress.com/feed/ to your RSS reader. If you need an RSS reader, sign up for a free account on Dreamwidth or Livejournal; you can put RSS feeds on your Friends page. Or if you prefer a more old-fashioned way that shows up in my statistics here, use the “Follow Nebusresearch” button at the upper right corner of this page and follow it through your WordPress account. Or follow my fallow @Nebusj account on Twitter where new posts get announced at least. As ever, thank you for reading. Be well, please.
There were a good number of comic strips mentioning mathematical subjects last week, as you might expect for one including the 14th of March. Most of them were casual mentions, though, so that’s why this essay looks like this. And is why the week will take two pieces to finish.
Paul Trap’s Thatababy for the 9th is a memorial strip to Katherine Johnson. She was, as described, a NASA mathematician, and one of the great number of African-American women whose work computing was rescued from obscurity by the book and movie Hidden Figures. NASA, and its associated agencies, do a lot of mathematical work. Much of it is numerical mathematics: a great many orbital questions, for example, can not be answered with, like, the sort of formula that describes how far away a projectile launched on a parabolic curve will land. Creating a numerical version of a problem requires insight and thought about how to represent what we would like to know. And calculating that requires further insight, so that the calculation can be done accurately and speedily. (I think about sometime doing a bit about the sorts of numerical computing featured in the movie, but I would hardly be the first.)
I also had thought the Mathematical Moments from the American Mathematical Society had posted an interview with her last year. I was mistaken but in, I think, a forgivable way. In the episode “Winning the Race”, posted the 12th of June, they interviewed Christine Darden, another of the people in the book, though not (really) the movie. Darden joined NASA in the late 60s. But the interview does talk about this sort of work, and how it evolved with technology. And, of course, mentions Johnson and her influence.
Stephen Beals’s Adult Children for the 11th has a character mourning that he took calculus as he’s “too stupid to be smart”. Knowing mathematics is often used as proof of intelligence. And calculus is used as the ultimate of mathematics. It’s a fair question why calculus and not some other field of mathematics, like differential equations or category theory or topology. Probably it’s a combination of slightly lucky choices (for calculus). Calculus is old enough to be respectable. It’s often taught as the ultimate mathematics course that people in high school or college (and who aren’t going into a mathematics field) will face. It’s a strange subject. Learning it requires a greater shift in thinking about how to solve problems than even learning algebra does. And the name is friendly enough, without the wordiness or technical-sounding language of, for example, differential equations. The subject may be well-situated.
I’ll have the rest of the week’s strips, including what Comic Strip Master Command ordered done for Pi Day, soon. And again I mention that I’m hosting this month’s Playful Math Education Blog Carnival. If you have come across a web site with some bit of mathematics that brought you delight and insight, please let me know, and mention any creative projects that you have, that I may mention that too. Thank you.
So the first bit of news: I’m hosting the Playful Math Education Blog Carnival later this month. This is a roaming blog link party, sharing blogs that delight or educate, or ideally both, about mathematics. As mentioned the other day Iva Sallay of Find the Factors hosted the 135th of these. My entry, the 136th, I plan to post sometime the last week of March.
And I’ll need help! If you’ve run across a web site, YouTube video, blog post, or essay that discusses something mathematical in a way that makes you grin, please let me know, and let me share it with the carnival audience.
This Saturday is March 14th, which we’ve been celebrating as Pi Day. I remain skeptical that it makes a big difference in people’s view of mathematics or in their education. But an afternoon spent talking about mathematics with everyone agreeing that, for today, we won’t complain about how hard it always was or how impossible we always found it, is pleasant. And that’s a good thing. I don’t know how much activity there’ll be for it, since the 14th is a weekend day this year. And the Covid-19 problem has got all the schools in my state closed through to April, so any calendar relevance is shattered.
But I have some things in the archive anyway. Last year I gathered Six Or Arguably Four Things For Pi Day, a collection of short essays about ways to calculate π well or poorly, and about some of the properties we’re pretty sure that π has, even if we can’t prove it. Also this fascinating physics problem that yields the digits of π.
And the middle of March often brings out Comic Strip Master Command. It looks like I’ve had at least five straight Pi Day editions of Reading the Comics, although most of them cover strips from more than just the 14th of March. From the past:
I apologize that obligations have kept me from writing some things that I mean to. So let me just point you to Iva Sallay, whose Find the Factor recreational math puzzle page hosted the 135th Playful Math Education Blog Carnival this past month. The Blog Carnival is a fun roaming thing that I’ve hosted once, and do hope to host again. It’s a curated collection of other mathematics sites that are fun or interesting or hopefully both together.
It was another pretty quiet week for mathematically-themed comic strips. Most of what did mention my subject just presented it as a subject giving them homework or quizzes or exams. But let’s look over what is here.
Ted Shearer’s Quincy for the 5th is the most interesting strip of the week, since it suggests an actual answerable mathematics problem. How much does a professional basketball player earn per dribble? The answer requires a fair bit of thought, like, what do you mean by “a professional basketball player”? There’s many basketball leagues around the world; even if we limit the question to United States-and-Canada leagues, there’s a fair number of minor leagues. If we limit it to the National Basketball Association there’s the question of whether the salary is the minimum union contract guarantee, or the mean salary, or the median salary. It’s exciting to look at the salary of the highest-paid players, too, of course.
Working out the number of dribbles per year is also a fun estimation challenge. Even if we pick a representative player there’s no getting an exact count of how many dribbles they’ve made over a year, even if we just consider “dribbling during games” to be what’s paid for. (And any reasonable person would have to count all the dribbling done during warm-up and practice as part of what’s being paid for.) But someone could come up with an estimate of, for example, about how long a typical player has the ball for a game, and how much of that time is spent moving the ball or preparing for a free throw or other move that calls for dribbling. How long a dribble typically takes. How many games a player typically plays over the year. The estimate you get from this will never, ever, be exactly right. But it should be close enough to give an idea how much money a player earns in the time it takes to dribble the ball once. So occasionally the comics put forth a good story problem after all.
Quincy on the 7th is again worrying about his mathematics and spelling tests. It’s a cute coincidence that these are the subjects worried about in Wee Pals too.
Paul Gilligan’s Pooch Cafe for the 7th is part of a string of jokes about famous dogs. This one’s a riff on Albert Einstein, mentioned here because Albert Einstein has such strong mathematical associations.
This week the BBC podcast In Our Time, a not-quite-hourlong panel show discussing varied topics, came to Paul Dirac. It can be heard here, or from other podcast sources. I get it off iTunes myself. The discussion is partly about his career and about the magnitude of his work. It’s not going to make anyone suddenly understand how to do any of his groundbreaking work in quantum mechanics. But it is, after all, an hourlong podcast for the general audience about, in this case, a physicist. It couldn’t explain spinors.
And even if you know a fair bit about Dirac and his work you might pick up something new. This might be slight: one of the panelists mentioned Dirac, in retirement, getting to know Sting. This is not something impossible, but it’s also not a meeting I would have ever imagined happening. So my week has been broadened a bit.
I do read other people’s mathematics writing, even if I don’t do it enough. A couple days ago RJ Lipton and KW Regan’s Reductions And Jokes discussed how one can take a problem and rewrite it as a different problem. This is one of the standard mathematician’s tricks. The point to doing this is that you might have a better handle on the new problem.
“Better” is an aesthetic judgement. It reflects whether the new problem is easier to work with. Along the way, they offer an example that surprised and delighted me, and that I wanted to share. It’s about multiplying whole numbers. Multiplication can take a fair while, as anyone who’s tried to do 38 times 23 by hand has found out. But we can speed that up. A multiplication table is a special case of a lookup table, a chunk of stored memory which has computed ahead of time all the multiplications someone is likely to do. Then instead of doing them, you just look them up.
The catch is that a multiplication table takes memory. To do all the multiplications for whole numbers 1 through 10 you need … well, not 100 memory cells. But 55. To have 1 through 20 worked out ahead of time you need 210 memory cells. Can we do better?
If addition and subtraction are easy enough to do? And if dividing by two is easy enough? Then, yes. Instead of working out every pair multiplication, work out the squares of the whole numbers. And then make use of this identity:
And that delights me. It’s one of those relationships that’s sitting there, waiting for anyone who’s ever squared a binomial to notice. I don’t know that anyone actually uses this. But it’s fun to see multiplication worked out by a different yet practical way.
I can clear out all last week’s mathematically-themed comic strips in one move, it looks like. There were a fair number of strips; it’s just they mostly mention mathematics in passing.
Bill Amend’s FoxTrot for the 23rd — a new strip; it’s still in original production for Sundays — has Jason asking his older sister to double-check a mathematics problem. Double-checking work is reliably useful, as proof against mistakes both stupid and subtle. But that’s true of any field.
Jim Unger’s Herman for the 23rd has a parent complaining about the weird New Math. The strip is a rerun and I don’t know from when; it hardly matters. The New Math has been a whipping boy for mathematics education since about ten minutes after its creation. And the complaint attaches to every bit of mathematics education reform ever. I am sympathetic to parents, who don’t see why their children should be the test subjects for a new pedagogy. And who don’t want to re-learn mathematics in order to understand what their children are doing. But, still, let someone know you were a mathematics major and they will tell you how much they didn’t understand or like mathematics in school. It’s hard to see why not try teaching it differently.
(If you do go out pretending to be a mathematics major, don’t worry. If someone challenges you on a thing, cite “Euler’s Theorem”, and you’ll have said something on point. And I’ll cover for you.)
Brian Gordon’s Fowl Language for the 25th has a father trying to explain the vastness of Big Numbers to their kid. Past a certain point none of us really know how big a thing is. We can talk about 300 sextillion stars, or anything else, and reason can tell us things about that number. But do we understand it? Like, can we visualize that many stars the way we can imagine twelve stars? This gets us into the philosophy of mathematics pretty soundly. 300 sextillion is no more imaginary than four is, but I know I feel more confident in my understanding of four. How does that make sense? And can you explain that to your kid?
Vic Lee’s Pardon my Planet for the 28th has an appearance by Albert Einstein. And a blackboard full of symbols. The symbols I can make out are more chemistry than mathematics, but they do exist just to serve as decoration.
Yesterday was the birthday of Herman Hollerith. His 40th since his birth in 1860. He’s renowned in computing circles. His work in automating the counting and of data made the United States’s 1890 Census possible. This is not the ordinary hyperbole: the 1880 Census’s data took eight years to fully collate. Hollerith’s tabulating machines took … well, six years for the full job, but they were keeping track of quite a bit of information. Hollerith’s system would go on to be used for other censuses, and also for general inventory and data-tracking purposes. His tabulating company would go on to be one of the original components of IBM. Cards, card readers, and card sorters with a clear lineage to this system would be used until fully electronic computers took over.
(It’s commonly assumed that the traditional 80-character width of a text terminal traces to the 80-hole punch cards which became the standard. Programmers particularly love to tell that tale, ignoring early computing screens that had different lengths, particularly 72 characters. More plausibly 80 characters owes to two things: it’s a nice round number, and it’s close to the number of characters you can type on a standard sheet of paper with a normal typewriter font. So it’s about the “right” length, one that we’ve been trained to accept as enough text to read at a glance.)
Well. In about 1970 IBM hired Bob Newhart to record a bit, for … fun, if that word applies to IBM. Part of the publicity for launching the famous System 370 machine. The structure echos the bit where Bob Newhart imagines being the first guy to hear of Sir Walter Raleigh’s importing of tobacco, and just how weird every bit of that is. In this bit, Newhart imagines talking on the phone with Herman Hollerith and hearing about just how this punched-card system is supposed to work. For decades, though, the film was reported lost.
What I did not know until mentioning to a friend two days ago is: the film was found! And a decade ago! In a Swedish bank vault because that’s the way this sort of thing always happens. Which is a neat bit of historical rhyming: the original fine data from the first Hollerith census of 1890 is lost, most likely destroyed in 1933 or 1934. So, please let me share with you Bob Newhart hearing about Herman Hollerith’s system. The end appears to be cut off, and there are Swedish subtitles that might just give away a couple jokes, if you can’t help paying attention to them.
Like a lot of comic work-for-hire it’s not Newhart’s best. It’s not going to displace the Voyage of the USS Codfish in my heart. There are a few spots to me where it seems like Newhart’s overlooked a good additional punch line, and I don’t know whether that reflects Newhart wanting to keep the piece from growing too long or too technical or what. It’s possible Newhart didn’t feel familiar enough with punch card technology to get too technical too. Newhart did work, briefly, as an accountant and might have had some reason to use the things. But I’m not aware of his telling any stories of doing so, and that seems a telling omission.
Still, it’s great to see this bit has been preserved, and is available. And is a Bob Newhart routine about early computer technologies, somehow.
Anyone hoping for an answer besides the 29th of February either suspects I’m doing some clickbait thing, maybe talking about that time Sweden didn’t quite make the transition from the Julian to the Gregorian calendar, or realizes I’m talking about days of the week. Are 29ths of February more likely to be a Sunday, a Monday, what?
Or, for that matter, if we don’t know which leap year we’re thinking of. There are 97 of them every 400 years. Since 97 things can’t be uniformly spread across the seven days of the week, how are they spread?
This is what computers are for. You’ve seen me do this for the date of Easter and for the date of (US) Thanksgiving. Using the ‘weekday’ function in Octave (a Matlab clone) I checked. In any 400-year span of the Gregorian calendar — and the calendar recycles every 400 years, so that’s as much as we need — we will see this distribution:
Leap Day will be a
this many times
in 400 years
Through to 2100, though, the calendar is going to follow a 28-year period. So this will be the last Saturday leap day until 2048. The next several ones will be Thursday, Tuesday, Sunday, Friday, Wednesday, and Monday.
So way back about fifty years ago, when pop science started to seriously explain how computers worked, and when the New Math fad underscored how much mathematics is an arbitrary cultural choice, the existence of number bases other than ten got some publicity. This offered the chance for a couple of jokes, or at least things which read to pop-science-fans as jokes. For example, playing on a typographical coincidence between how some numbers are represented in octal (base eight) and decimal (base ten), we could put forth this: for computer programmers Halloween is basically another Christmas. After all, 31 OCT = 25 DEC. It’s not much of a joke, but how much of a joke could you possibly make from “writing numbers in different bases”? Anyway, Isaac Asimov was able to make a short mystery out of it.
Tony Cochrane’s Agnes for the 21st is part of a sequence with Agnes having found some manner of tablet computer. Automatic calculation has always been a problem in teaching arithmetic. A computer’s always able to do more calculations, more accurately, than a person is; so, whey do people need to learn anything about how to calculate? The excuse that we might not always have a calculator was at least a little tenable up to about fifteen years ago. Now it’d take a massive breakdown in society for computing devices not to be pretty well available. This would probably take long enough for us to brush up on long division.
It’s more defensible to say that people need to be able to say whether an answer is plausible. If we don’t have any expectations for the answer, we don’t know whether we’ve gone off and calculated a wrong thing. This is a bit more convincing. We should have some idea whether 25, 2500, or 25 million is the more likely answer. That won’t help us spot whether we made a mistake and got 27 instead of 25, though. It does seem reasonable to say that we can’t appreciate mathematics, so much of which is studying patterns and structures, without practicing. And arithmetic offers great patterns and structures, while still being about things that we find familiar and useful. So that’s likely to stay around.
John Rose’s Barney Google and Snuffy Smith for the 21st is a student-subverting-the-blackboard-problem joke. Jughaid’s put the arithmetic problems into terms of what he finds most interesting. To me, it seems like if this is helping him get comfortable with the calculations, let him. If he does this kind of problem often enough, he’ll get good at it and let the false work of going through sports problems fade away.
Stephan Pastis’s Pearls Before Swine for the 21st sees Pig working through a simple Retirement Calculator. He appreciates the mathematics being easy. A realistic model would have wrinkles to it. For example, the retirement savings would presumably be returning interest, from investments or from simple deposit accounts. Working out how much one gets from that, combined with possibly spending down the principal, can be involved. But a rough model doesn’t need this sort of detailed complication. It can be pretty simple, and still give you some guidance to what a real answer should look like.
John Zakour and Scott Roberts’s Working Daze for the 21st is a joke about how guys assuming that stuff they like is inherently interesting to other people. In this case, it’s hexadecimal arithmetic. That’s at least got the slight appeal that we’ve settled on using a couple of letters as numerals for it, so that wordplay and word-like play is easier than it is in base ten.
And this wraps up a string of comic strips all with some mathematical theme that all posted on the same day. I grant none of these get very deep into mathematical topics; that’s all right. There’ll be some more next week in a post at this link. Thank you.