I had remembered this comic strip, and I hoped to use it for yesterday’s A-to-Z essay about Imaginary Numbers. But I wasn’t able to find it before publishing deadline. I figured I could go back and add this to the essay once I found it, and I likely will anyway. (The essay is quite long and any kind of visual appeal helps.)
But I also wanted folks to have the chance to notice it, and an after-the-fact addition doesn’t give that chance.
It is almost certain that Bill Watterson read this strip, and long before his own comic with eleventeen and thirty-twelve and such. Watterson has spoken of Schulz’s influence. That isn’t to say that he copied the joke. “Gibberish number-like words” is not a unique idea, and it’s certainly not original to Schulz. I’d imagine a bit of effort could find prior examples even within comic strips. (I’m reminded in Pogo of Howland Owl describing the Groundhog Child’s gibberish as first-rate algebra.) It’s just fun to see great creative minds working out similar ideas, and how they use those ideas for different jokes.
I have another topic today suggested by Beth, of the I Didn’t Have My Glasses On …. inspiration blog. It overlaps a bit with other essays I’ve posted this A-to-Z sequence, but that’s all right. We get a better understanding of things by considering them from several perspectives. This one will be a bit more historical.
Pop science writer Isaac Asimov told a story he was proud of about his undergraduate days. A friend’s philosophy professor held court after class. One day he declared mathematicians were mystics, believing in things they even admit are “imaginary numbers”. Young Asimov, taking offense, offered to prove the reality of the square root of minus one, if the professor gave him one-half pieces of chalk. The professor snapped a piece of chalk in half and gave one piece to him. Asimov said this is one piece of chalk. The professor answered it was half the length of a piece of chalk and Asimov said that’s not what he asked for. Even if we accept “half the length” is okay, how do we know this isn’t 48 percent the length of a standard piece of chalk? If the professor was that bad on “one-half” how could he have opinions on “imaginary numbers”?
This story is another “STEM undergraduates outwitting the philosophy expert” legend. (Even if it did happen. What we know is the story Asimov spun it into, in which a plucky young science fiction fan out-argued someone whose job is forming arguments.) Richard Feynman tells a similar story, befuddling a philosophy class with the question of how we can prove a brick has a interior. It helps young mathematicians and science majors feel better about their knowledge. But Asimov’s story does get at a couple points. First, that “imaginary” is a terrible name for a class of numbers. The square root of minus one is as “real” as one-half is. Second, we’ve decided that one-half is “real” in some way. What the philosophy professor would have baffled Asimov to explain is: in what way is one-half real? Or minus one?
We’re introduced to imaginary numbers through polynomials. I mean in education. It’s usually right after getting into quadratics, looking for solutions to equations like . That quadratic has two solutions, but it’s possible to have a quadratic with only one, such as . Or to have a quadratic with no solutions, such as, iconically, . We might underscore that by plotting the curve whose x- and y-coordinates makes true the equation . There’s no point on the curve with a y-coordinate of zero, so, there we go.
Having established that has no solutions, the course then asks “what if we go ahead and say there was one”? Two solutions, in fact, and . This is all right for introducing the idea that mathematics is a tool. If it doesn’t do something we need, we can alter it.
But I see trouble in teaching someone how you can’t take square roots of negative numbers and then teaching them how to take square roots of negative numbers. It’s confusing at least. It needs some explanation about what changed. We might do better introducing them in a more historical method.
Historically, imaginary numbers (in the West) come from polynomials, yes. Different polynomials. Cubics, and quartics. Mathematicians still liked finding roots of them. Mathematicians would challenge one another to solve sets of polynomials. This seems hard to believe, but many sources agree on this. I hope we’re not all copying Eric Temple Bell here. (Bell’s Men of Mathematics is an inspiring collection of biographical sketches. But it’s not careful differentiating legends from documented facts.) And there are enough nerd challenges today that I can accept people daring one another to find solutions of .
Quadratics, equations we can write as for some real numbers a, b, and c, we’ve known about forever. Euclid solved these kinds of equations using geometric reasoning. Chinese mathematicians 2200 years ago described rules for how to find roots. The Indian mathematician Brahmagupta, by the early 7th century, described the quadratic formula to find at least one root. Both possible roots were known to Indian mathematicians a thousand years ago. We’ve reduced the formula today to
With that filtering into Western Europe, the search was on for similar formulas for other polynomials. This turns into several interesting threads. One is a tale of intrigue and treachery involving Gerolamo Cardano, Niccolò Tartaglia, and Ludovico Ferrari. I’ll save that for another essay because I have to cut something out, so of course I skip the dramatic thing. Another thread is the search for quadratic-like formulas for other polynomials. They exist for third-power and fourth-power polynomials. Not (generally) for the fifth- or higher-powers. That is, there are individual polynomials you can solve by formulas, like, . But stare at it and you can see where that’s “really” a quadratic pretending to be sixth-power. Finding there was no formula to find, though, lead people to develop group theory. And group theory underlies much of mathematics and modern physics.
The first great breakthrough solving the general cubic, , came near the end of the 14th century in some manuscripts out of Florence. It’s built on a transformation. Transformations are key to mathematics. The point of a transformation is to turn a problem you don’t know how to do into one you do. As I write this, MathWorld lists 543 pages as matching “transformation”. That’s about half what “polynomial” matches (1,199) and about three times “trigonometric” (184). So that can help you judge importance.
Here, the transformation to make is to write a related polynomial in terms of a new variable. You can call that new variable x’ if you like, or z. I’ll use z so as to not have too many superscript marks flying around. This will be a “depressed polynomial”. “Depressed” here means that at least one of the coefficients in the new polynomial is zero. (Here, for this problem, it means we won’t have a squared term in the new polynomial.) I suspect the term is old-fashioned.
Let z be the new variable, related to x by the equation . And then figure out what and are. Using all that, and the knowledge that , and a lot of arithmetic, you get to one of these three equations:
where p and q are some new coefficients. They’re positive numbers, or possibly zeros. They’re both derived from a, b, c, and d. And so in the 15th Century the search was on to solve one or more of these equations.
From our perspective in the 21st century, our first question is: what three equations? How are these not all the same equation? And today, yes, we would write this as one depressed equation, most likely . We would allow that p or q or both might be negative numbers.
And there is part of the great mysterious historical development. These days we generally learn about negative numbers. Once we are comfortable, our teachers hope, with those we get imaginary numbers. But in the Western tradition mathematicians noticed both, and approached both, at roughly the same time. With roughly similar doubts, too. It’s easy to point to three apples; who can point to “minus three” apples? We can arrange nine apples into a neat square. How big a square can we set “minus nine” apples in?
Hesitation and uncertainty about negative numbers would continue quite a long while. At least among Western mathematicians. Indian mathematicians seem to have been more comfortable with them sooner. And merchants, who could model a negative number as a debt, seem to have gotten the idea better.
But even seemingly simple questions could be challenging. John Wallis, in the 17th century, postulated that negative numbers were larger than infinity. Leonhard Euler seems to have agreed. (The notion may seem odd. It has echoes today, though. Computers store numbers as bit patterns. The normal scheme represents negative numbers by making the first bit in a pattern 1. These bit patterns make the negative numbers look bigger than the biggest positive numbers. And thermodynamics gives us a temperature defined by the relationship of energy to entropy. That definition implies there can be negative temperatures. Those are “hotter” — higher-energy, at least — than infinitely-high positive temperatures.) In the 18th century we see temperature scales designed so that the weather won’t give negative numbers too often. Augustus De Morgan wrote in 1831 that a negative number “occurring as the solution of a problem indicates some inconsistency or absurdity”. De Morgan was not an amateur. He coded the rules for deductive logic so well we still call them De Morgan’s laws. He put induction on a logical footing. And he found negative numbers (and imaginary numbers) a sign of defective work. In 1831. 1831!
But back to cubic equations. Allow that we’ve gotten comfortable enough with negative numbers we only want to solve the one depressed equation of . How to do it? … Another transformation, then. There are a couple you can do. Modern mathematicians would likely define a new variable w, set so that . This turns the depressed equation into
And this, believe it or not, is a disguised quadratic. Multiply everything in it by and move things around a little. You get
From there, quadratic formula to solve for . Then from that, take cube roots and you get three values of z. From that, you get your three values of x.
You see why nobody has taught this in high school algebra since 1959. Also why I am not touching the quartic formula, the equivalent of this for polynomials of degree four.
There are other approaches. And they can work out easier for particular problems. Take, for example, which I introduced in the first act. It’s past the time we set it off.
Rafael Bombelli, in the 1570s, pondered this particular equation. Notice it’s already depressed. A formula developed by Cardano addressed this, in the form . Notice that’s the second of the three sorts of depressed polynomial. Cardano’s formula says that one of the roots will be at
Put to this problem, we get something that looks like a compelling reason to stop:
Bombelli did not stop with that, though. He carried on as though these expressions of the square root of -121 made sense. And, if he did that he found these terms added up. You get an x of 4.
Which is true. It’s easy to check that it’s right. And here is the great surprising thing. Start from the respectable enough equation. It has nothing suspicious in it, not even negative numbers. Follow it through and you need to use negative numbers. Worse, you need to use the square roots of negative numbers. But keep going, as though you were confident in this, and you get a correct answer. And a real number.
We can get the other roots. Divide out of . What’s left is . You can use the quadratic formula for this. The other two roots are , about -0.268, and , about -3.732.
So here we have good reasons to work with negative numbers, and with imaginary numbers. We may not trust them. But they get us to correct answers. And this brings up another little secret of mathematics. If all you care about is an answer, then it’s all right to use a dubious method to get an answer.
There is a logical rigor missing in “we got away with it, I guess”. The name “imaginary numbers” tells of the disapproval of its users. We get the name from René Descartes, who was more generally discussing complex numbers. He wrote something like “in many cases no quantity exists which corresponds to what one imagines”.
John Wallis, taking a break from negative numbers and his other projects and quarrels, thought of how to represent imaginary numbers as branches off a number line. It’s a good scheme that nobody noticed at the time. Leonhard Euler envisioned matching complex numbers with points on the plane, but didn’t work out a logical basis for this. In 1797 Caspar Wessel presented a paper that described using vectors to represent complex numbers. It’s a good approach. Unfortunately that paper too sank without a trace, undiscovered for a century.
In 1806 Jean-Robert Argand wrote an “Essay on the Geometrical Interpretation of Imaginary Quantities”. Jacques Français got a copy, and published a paper describing the basics of complex numbers. He credited the essay, but noted that there was no author on the title page and asked the author to identify himself. Argand did. We started to get some good rigor behind the concept.
In 1831 William Rowan Hamilton, of Hamiltonian fame, described complex numbers using ordered pairs. Once we can define their arithmetic using the arithmetic of real numbers we have a second solid basis. More reason to trust them. Augustin-Louis Cauchy, who proved about four billion theorems of complex analysis, published a new construction of them. This used a group theory approach, a polynomial ring we denote as . I don’t have the strength to explain all that today. Matrices give us another approach. This matches complex numbers with particular two-row, two-column matrices. This turns the addition and multiplication of numbers into what Hamilton described.
And here we have some idea why mathematicians use negative numbers, and trust imaginary numbers. We are pushed toward them by convenience. Negative numbers let us work with one equation, , rather than three. (Or more than three equations, if we have to work with an x we know to be negative.) Imaginary numbers we can start with, and find answers we know to be true. And this encourages us to find reasons to trust the results. Having one line of reasoning is good. Having several lines — Argand’s geometric, Hamilton’s coordinates, Cauchy’s rings — is reassuring. We may not be able to point to an imaginary number of anything. But if we can trust our arithmetic on real numbers we can trust our arithmetic on imaginary numbers.
As I mentioned Descartes gave the name “imaginary number” to all of what we would now call “complex numbers”. Gauss published a geometric interpretation of complex numbers in 1831. And gave us the term “complex number”. Along the way he complained about the terminology, though. He noted “had +1, -1, and , instead of being called positive, negative, and imaginary (or worse still, impossible) unity, been given the names say, of direct, inverse, and lateral unity, there would hardly have been any scope for such obscurity”. I’ve never heard them term “impossible numbers”, except as an adjective.
The name of a thing doesn’t affect what it is. It can affect how we think about it, though. We can ask whether Asimov’s professor would dismiss “lateral numbers” as mysticism. Or at least as more mystical than “three” is. We can, in context, understand why Descartes thought of these as “imaginary numbers”. He saw them as something to use for the length of a calculation, and that would disappear once its use was done. We still have such concepts, things like “dummy variables” in a calculus problem. We can’t think of a use for dummy variables except to let a calculation proceed. But perhaps we’ll see things differently in four hundred years. Shall have to come back and check.
This was a week of few mathematically-themed comic strips. I don’t mind. If there was a recurring motif, it was about parents not doing mathematics well, or maybe at all. That’s not a very deep observation, though. Let’s look at what is here.
Liniers’s Macanudo for the 18th puts forth 2020 as “the year most kids realized their parents can’t do math”. Which may be so; if you haven’t had cause to do (say) long division in a while then remembering just how to do it is a chore. This trouble is not unique to mathematics, though. Several decades out of regular practice they likely also have trouble remembering what the 11th Amendment to the US Constitution is for, or what the rule is about using “lie” versus “lay”. Some regular practice would correct that, though. In most cases anyway; my experience suggests I cannot possibly learn the rule about “lie” versus “lay”. I’m also shaky on “set” as a verb.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 18th shows a mathematician talking, in the jargon of first and second derivatives, to support the claim there’ll never be a mathematician president. Yes, Weinersmith is aware that James Garfield, 20th President of the United States, is famous in trivia circles for having an original proof of the Pythagorean theorem. It would be a stretch to declare Garfield a mathematician, though, except in the way that anyone capable of reason can be a mathematician. Raymond Poincaré, President of France for most of the 1910s and prime minister before and after that, was not a mathematician. He was cousin to Henri Poincaré, who founded so much of our understanding of dynamical systems and of modern geometry. I do not offhand know what presidents (or prime ministers) of other countries have been like.
Weinersmith’s mathematician uses the jargon of the profession. Specifically that of calculus. It’s unlikely to communicate well with the population. The message is an ordinary one, though. The first derivative of something with respect to time means the rate at which things are changing. The first derivative of a thing, with respect to time being positive means that the quantity of the thing is growing. So, that first half means “things are getting more bad”.
The second derivative of a thing with respect to time, though … this is interesting. The second derivative is the same thing as the first derivative with respect to time of “the first derivative with respect to time”. It’s what the change is in the rate-of-change. If that second derivative is negative, then the first derivative will, in time, change from being positive to being negative. So the rate of increase of the original thing will, in time, go from a positive to a negative number. And so the quantity will eventually decline.
So the mathematician is making a this-is-the-end-of-the-beginning speech. The point at which the the second derivative of a quantity changes sign is known as the “inflection point”. Reaching that is often seen as the first important step in, for example, disease epidemics. It is usually the first good news, the promise that there will be a limit to the badness. It’s also sometimes mentioned in economic crises or sometimes demographic trends. “Inflection point” is likely as technical a term as one can expect the general public to tolerate, though. Even that may be pushing things.
Julie Larson’s The Dinette Set rerun for the 21st fusses around words. Along the way Burl mentions his having learned that two negatives can make a positive, in mathematics. Here it’s (most likely) the way that multiplying or dividing two negative numbers will produce a positive number.
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.
There were a bunch of comic strips mentioning some kind of mathematical theme last week. I need to clear some out. So I’ll start with some of the marginal mentions. Many of these involve having to deal with exams or quizzes.
There are different ways to find square roots. (I can guarantee that Skip wasn’t expected to use this one.) The term ‘root’ derives from an idea that the root of a number is the thing that generates it: 3 is a square root of 9 because multiplying 3’s together gives you 9. ‘Square’ is I have always only assumed because multiplying a number by itself will give you the area of a square with sides of length that number. This is such an obvious word origin, though, that I am reflexively suspicious. Word histories are usually subtle and capricious things.
The strip for the 8th closing the storyline has a nice example of using “billion” as a number so big as to be magical, capable of anything. Big numbers can do strange and contrary-to-intuition things. But they can be reasoned out.
Tony Cochran’s Agnes for the 4th sees the title character figuring she could sell her “personal smartness”. Her best friend Trout wonders if that’s tutoring math or something. (Incidentally, Agnes is one of the small handful of strips to capture what made Calvin and Hobbes great; I recommend giving it a try.)
Charles Schulz’s Peanuts Begins for the 5th sees Charlie Brown working problems on the board. He’s stuck for what to do until he recasts the problem as scoring in football and golf. We may giggle at this, but I support his method. It’s convinced him the questions are worth solving, the most important thing to doing them at all. And it’s gotten him to the correct answers. Casting these questions as sports problems is the building of falsework: it helps one do the task, and then is taken away (or hidden) from the final product. Everyone who does mathematics builds some falsework like this. If we do a particular problem, or kind of problem, often enough we get comfortable enough with the main work that we don’t need the falsework anymore. So it is likely to be for Charlie Brown.
I like this scheme where I use the Sunday publication slot to list comics that mention mathematics without inspiring conversation. I may need a better name for that branch of the series, though. But, nevertheless, here are comic strips from last week that don’t need much said about them.
John Deering’s Strange Brew for the 24th features Pythagoras, here being asked about his angles. I’m not aware of anything actually called a Pythagorean Angle, but there’s enough geometric things with Pythagoras’s name attached for the joke to make sense.
Maria Scrivan’s Half Full for the 25th is a Venn Diagram joke for the week. It doesn’t quite make sense as a Venn Diagram, as it’s not clear to me that “invasive questions” is sensibly a part of “food”. But it’s a break from every comic strip doing a week full of jokes about turkeys preferring to not be killed.
Tony Carrillo’s F Minus for the 26th is set in mathematics class. And talks about how the process of teaching mathematics is “an important step on the road to hating math”, which is funny because it’s painfully true.
As will sometimes happen it’s inconvenient for met to write up a paragraph or two on the particularly mathematically significant comic strips of the past week. Let me here share the comics that just mentioned mathematics, then, and save the heavy stuff for a bit later on.
And this covers things through to Friday’s comics. I write this not having had the chance to read Saturday’s yet. When I do, and when I have the whole week’s strips to discuss, I’ll have it at this link. Furthermore, this week sees the last quarter of the Fall 2019 A to Z under way. I’m excited to learn what I’m doing for the letter ‘U’ also.
If there is a theme to the last comic strips from the previous week, it’s that kids find arithmetic hard. That’s a title for you.
Bill Watterson’s Calvin and Hobbes for the 2nd is one of the classics, of course. Calvin’s made the mistake of supposing that mathematics is only about getting true answers. We’ll accept the merely true, if that’s what we can get. But we want interesting. Which is stuff that’s not just true but is unexpected or unforeseeable in some way. We see this when we talk about finding a “proper” answer, or subset, or divisor, or whatever. Some things are true for every question, and so, who cares?
Also, is it really true that Calvin doesn’t know any of his homework problems? It’s possible, but did he check?
Were I grading, I would accept an “I don’t know”, at least for partial credit, in certain conditions. Those involve the student writing out what they would like to do to try to solve the problem. If the student has a fair idea of something that ought to find a correct answer, then the student’s showing some mathematical understanding. But there are times that what’s being tested is proficiency at an operation, and a blank “I don’t know” would not help much with that.
Patrick Roberts’s Todd the Dinosaur for the 2nd has an arithmetic cameo. Fractions, particularly. They’re mentioned as something too dull to stay awake through. So for the joke’s purpose this could have been any subject that has an exposition-heavy segment. Fractions do have more complicated rules than adding whole numbers do. And introducing those rules can be hard. But anything where you introduce rules instead of showing what you can do with them is hard. I’m thinking here of several times people have tried to teach me board games by listing all the rules, instead of setting things up and letting me ask “what am I allowed to do now?” the first couple turns. I’m not sure how that would translate to fractions, but there might be something.
John Zakour and Scott Roberts’s Maria’s Day for the 2nd has another of Maria’s struggles with arithmetic. It’s presented as a challenge so fierce it can defeat even superheroes. Could be any subject, really. It’s hard to beat the visual economy of having it be a division problem, though.
Rick Kirkman and Jerry Scott’s Baby Blues for the 3rd shows a bit of youthful enthusiasm. Hammie’s parents would rather that enthusiasm be put to memorizing multiplication facts. I’m not sure this would match the fun of building stuff. But I remember finding patterns inside the multiplication table fascinating. Like how you could start from a perfect square and get the same sequence of numbers as you moved out along a diagonal. Or tracing out where the same number appeared in different rows and columns, like how just everything could multiply into 24. Might be worth playing with some.
I had thought I’d culled some more pieces from my Twitter and other mathematics-writing-reading the last couple weeks and I’m not sure where it all went. I think I might be baffled by the repostings of things on Quanta Magazine (which has a lot of good mathematics articles, but not, like, a 3,000-word piece every day, and they showcase their archive just as anyone ought).
It reviews Kim Plofker’s 2008 text Mathematics In India, a subject that I both know is important — I love to teach with historic context included — and something that I very much bluff my way through. I mean, I do research things I expect I’ll mention, but I don’t learn enough of the big picture and a determined questioner could prove how fragile my knowledge was. So Plofker’s book should go on my reading list at least.
We all know Newton and Leibniz introduced Calculus in the 17th century, but Cauchy made seminal contributions to its precision in the 1800s, thus we might say Cauchy introduced what we now call Analysis. Here's some notes on 19th century (real) Analysis. https://t.co/crP9tyiELC
These are lecture notes about analysis. In the 19th century mathematicians tried to tighten up exactly what we meant by things like “functions” and “limits” and “integrals” and “numbers” and all that. It was a lot of good solid argument, and a lot of surprising, intuition-defying results. This isn’t something that a lay reader’s likely to appreciate, and I’m sorry for that, but if you do know the difference between Riemann and Lebesgue integrals the notes are likely to help.
And this, Daniel Grieser and Svenja Maronna’s Hearing The Shape Of A Triangle, follows up on a classic mathematics paper, Mark Kac’s Can One Hear The Shape Of A Drum? This is part of a class of problems in which you try to reconstruct what kinds of things can produce a signal. It turns out to be impossible to perfectly say what shape and material of a drum produced a certain sound of a drum. But. A triangle — the instrument, that is, but also the shape — has a simpler structure. Could we go from the way a triangle sounds to knowing what it looks like?
Back to my usual self-preening. January 2018 was a successful month around here, in terms of people reading stuff I write. According to WordPress, there were some 1,274 pages viewed from 670 unique visitors. That’s the largest number of pages viewed since March and April 2016, when I had a particularly successful A To Z going. It’s the greatest number of unique visitors since September 2017 when I had a less successful but still pretty good A To Z going. The page views were well above December 2017’s 899, and November’s 1,052. The unique visitors were well above December’s 599 and November’s 604.
Still, this popularity was accompanied by readers actually liking stuff. There were 112 pages liked in January, beating out the 71 in December and 70 in November by literally dozens of clicks. It’s the highest count since August of 2017 and summer’s A To Z sequence. There were more comments, too, 39 of them. December saw 24 and November 28 and, you see this coming, that’s the largest number of comments since summer 2017’s A To Z sequence.
The popular articles for January were two of the ones I expected, one of the Reading the Comics posts, and then two surprises. What were they? These.
Yes, it’s clickbait-y to talk about weird tricks for limits that mathematicians use. In my defense: mathematicians really do rely on these tricks all the time. So if it’s getting people stuff that’s useful then my conscience is as clear as it is for asking “How many grooves are on a record’s side?” and (implicitly) “How many kinds of trapezoid are there?”
If I’m counting right there were 50 countries from which I drew readers, if “European Union” counts as a country and if “Trinidad and Tobago” don’t count as two. Plus there’s Hong Kong and when you get down to it, “country” is a hard concept to pin down exactly. There were 14 single-reader countries. Here’s the roster of them all:
Hong Kong SAR China
Trinidad & Tobago
There were 53 countries sending me readers in December and 56 in November so I guess I’m concentrating? There were 15 single-reader countries in December and 22 in November. Russia and Uruguay were single-reader countries in December; Mexico’s been a single-reader country for three months now.
WordPress’s Insights panel says I started the month with 57,592 page views recorded, from 27,161 recorded unique visitors. It also shares with me the interesting statistics that, as I write this and before I post it, I’ve written 16 total posts this year, which have drawn an average two comments and seven likes per post. There’ve been 900 words per post, on average. Overall this year I’ve gotten 39 comments, 110 likes, and have published 14,398 words. I don’t know whether that counts image captions. But this also leads me to learn what previous year statistics were like; I’ve been averaging over 900 words per post since 2015. In 2015 I averaged about 750 words per post, and got three times as many likes and about twice as many comments per post. I’m sure that doesn’t teach me anything. At the least I won’t learn from it.
The last couple days of last week saw a rush of comics, although most of them were simpler things to describe. Bits of play on words, if you like.
Samson’s Dark Side of the Horse for the 4th of January, 2018, is one that plays on various meanings of “average”. The mean, alluded to in the first panel, is the average most people think of first. Where you have a bunch of values representing instances of something, add up the values, and divide by the number of instances. (Properly that’s the arithmetic mean. There’s some others, such as the geometric mean, but if someone’s going to use one of those they give you clear warning.) The median, in the second, is the midpoint, the number that half of all instances are less than. So you see the joke. If the distribution of intelligence is normal — which is a technical term, although it does mean “not freakish” — then the median and the mean should be equal. If you had infinitely many instances, and they were normally distributed, the two would be equal. With finitely many instances, the mean and the median won’t be exactly in line, for the same reason if you fairly toss a coin two million times it won’t turn up heads exactly one million times.
Dark Side of the Horse for the 5th delivers the Roman numerals joke of the year. And I did have to think about whether ‘D’ is a legitimate Roman numeral. This would be easier to remember before 1900.
Johnny Hart’s Back to BC for the 5th goes to the desire to quantify and count things. And to double-check what other people tell you about this counting. It’s easy, today, to think of the desire to quantify things as natural to humans. I’m not confident that it is. The history of statistics shows this gradual increase in the number and variety of things getting tracked. This strip originally ran the 11th of July, 1960.
Bill Watterson’s Calvin and Hobbes for the 5th talks about averages again. And what a population average means for individuals. It doesn’t mean much. The glory of statistics is that groups are predictable in a way that individuals are not.
Mark O’Hare’s Citizen Dog rerun for the 6th is part of a sequence in which Fergus takes a (human) child’s place in school. Mathematics gets used as a subject that’s just a big pile of unfamiliar terms if you just jump right in. Most subjects are like this if you take them seriously, of course. But mathematics has got an economy of technical terms to stuff into people’s heads, and that have to be understood to make any progress. In grad school my functional analysis professor took great mercy on us, and started each class with re-writing the definitions of all the technical terms introduced the previous class. Also of terms that might be a bit older, but that are important to get right, which is why I got through it confident I knew what a Sobolev Space was. (It’s a collection of functions that have enough derivatives to do your differential equations problem.) Numerator and denominator, we’re experts on by now.
There were a good number of mathematically-themed comic strips in the syndicated comics last week. Those from the first part of the week gave me topics I could really sink my rhetorical teeth into, too. So I’m going to lop those off into the first essay for last week and circle around to the other comics later on.
Jef Mallett’s Frazz started a week of calendar talk on the 31st of December. I’ve usually counted that as mathematical enough to mention here. The 1st of January as we know it derives, as best I can figure, from the 1st of January as Julius Caesar established for 45 BCE. This was the first Roman calendar to run basically automatically. Its length was quite close to the solar year’s length. It had leap days added according to a rule that should have been easy enough to understand (one day every fourth year). Before then the Roman calendar year was far enough off the solar year that they had to be kept in synch by interventions. Mostly, by that time, adding a short extra month to put things more nearly right. This had gotten all confusingly messed up and Caesar took the chance to set things right, running 46 BCE to 445 days long.
But why 445 and not, say, 443 or 457? And I find on research that my recollection might not be right. That is, I recall that the plan was to set the 1st of January, Reformed, to the first new moon after the winter solstice. A choice that makes sense only for that one year, but, where to set the 1st is literally arbitrary. While that apparently passes astronomical muster (the new moon as seen from Rome then would be just after midnight the 2nd of January, but hitting the night of 1/2 January is good enough), there’s apparently dispute about whether that was the objective. It might have been to set the winter solstice to the 25th of December. Or it might have been that the extra days matched neatly the length of two intercalated months that by rights should have gone into earlier years. It’s a good reminder of the difficulty of reading motivation.
Brian Fies’s The Last Mechanical Monster for the 1st of January, 2018, continues his story about the mad scientist from the Fleischer studios’ first Superman cartoon, back in 1941. In this panel he’s describing how he realized, over the course of his long prison sentence, that his intelligence was fading with age. He uses the ability to do arithmetic in his head as proof of that. These types never try naming, like, rulers of the Byzantine Empire. Anyway, to calculate the cube root of 50,653 in his head? As he used to be able to do? … guh. It’s not the sort of mental arithmetic that I find fun.
But I could think of a couple ways to do it. The one I’d use is based on a technique called Newton-Raphson iteration that can often be used to find where a function’s value is zero. Raphson here is Joseph Raphson, a late 17th century English mathematician known for the Newton-Raphson method. Newton is that falling-apples fellow. It’s an iterative scheme because you start with a guess about what the answer would be, and do calculations to make the answer better. I don’t say this is the best method, but it’s the one that demands me remember the least stuff to re-generate the algorithm. And it’ll work for any positive number ‘A’ and any root, to the ‘n’-th power.
So you want the n-th root of ‘A’. Start with your current guess about what this root is. (If you have no idea, try ‘1’ or ‘A’.) Call that guess ‘x’. Then work out this number:
Ta-da! You have, probably, now a better guess of the n-th root of ‘A’. If you want a better guess yet, take the result you just got and call that ‘x’, and go back calculating that again. Stop when you feel like your answer is good enough. This is going to be tedious but, hey, if you’re serving a prison term of the length of US copyright you’ve got time. (It’s possible with this sort of iterator to get a worse approximation, although I don’t think that happens with n-th root process. Most of the time, a couple more iterations will get you back on track.)
But that’s work. Can we think instead? Now, most n-th roots of whole numbers aren’t going to be whole numbers. Most integers aren’t perfect powers of some other integer. If you think 50,653 is a perfect cube of something, though, you can say some things about it. For one, it’s going to have to be a two-digit number. 103 is 1,000; 1003 is 1,000,000. The second digit has to be a 7. 73 is 343. The cube of any number ending in 7 has to end in 3. There’s not another number from 1 to 9 with a cube that ends in 3. That’s one of those things you learn from playing with arithmetic. (A number ending in 1 cubes to something ending in 1. A number ending in 2 cubes to something ending in 8. And so on.)
So the cube root has to be one of 17, 27, 37, 47, 57, 67, 77, 87, or 97. Again, if 50,653 is a perfect cube. And we can do better than saying it’s merely one of those nine possibilities. 40 times 40 times 40 is 64,000. This means, first, that 47 and up are definitely too large. But it also means that 40 is just a little more than the cube root of 50,653. So, if 50,653 is a perfect cube, then it’s most likely going to be the cube of 37.
Bill Watterson’s Calvin and Hobbes rerun for the 2nd is a great sequence of Hobbes explaining arithmetic to Calvin. There is nothing which could be added to Hobbes’s explanation of 3 + 8 which would make it better. I will modify Hobbes’s explanation of what the numerator. It’s ridiculous to think it’s Latin for “number eighter”. The reality is possibly more ridiculous, as it means “a numberer”. Apparently it derives from “numeratus”, meaning, “to number”. The “denominator” comes from “de nomen”, as in “name”. So, you know, “the thing that’s named”. Which does show the terms mean something. A poet could turn “numerator over denominator” into “the number of parts of the thing we name”, or something near enough that.
Hobbes continues the next day, introducing Calvin to imaginary numbers. The term “imaginary numbers” tells us their history: they looked, when first noticed in formulas for finding roots of third- and fourth-degree polynomials, like obvious nonsense. But if you carry on, following the rules as best you can, that nonsense would often shake out and you’d get back to normal numbers again. And as generations of mathematicians grew up realizing these acted like numbers we started to ask: well, how is an imaginary number any less real than, oh, the square root of six?
Hobbes’s particular examples of imaginary numbers — “eleventeen” and “thirty-twelve” — are great-sounding compositions. They put me in mind, as many of Watterson’s best words do, of a 1960s Peanuts in which Charlie Brown is trying to help Sally practice arithmetic. (I can’t find it online, as that meme with edited text about Sally Brown and the sixty grapefruits confounds my web searches.) She offers suggestions like “eleventy-Q” and asks if she’s close, which Charlie Brown admits is hard to say.
And finally, James Allen’s Mark Trail for the 3rd just mentions mathematics as the subject that Rusty Trail is going to have to do some work on instead of allowing the experience of a family trip to Mexico to count. This is of extremely marginal relevance, but it lets me include a picture of a comic strip, and I always like getting to do that.
I went a little wild explaining the first of last week’s mathematically-themed comic strips. So let me split the week between the strips that I know to have been reruns and the ones I’m not so sure were.
Bill Amend’s FoxTrot for the 23rd — not a rerun; the strip is still new on Sundays — is a probability question. And a joke about story problems with relevance. Anyway, the question uses the binomial distribution. I know that because the question is about doing a bunch of things, homework questions, each of which can turn out one of two ways, right or wrong. It’s supposed to be equally likely to get the question right or wrong. It’s a little tedious but not hard to work out the chance of getting exactly six problems right, or exactly seven, or exactly eight, or so on. To work out the chance of getting six or more questions right — the problem given — there’s two ways to go about it.
One is the conceptually easy but tedious way. Work out the chance of getting exactly six questions right. Work out the chance of getting exactly seven questions right. Exactly eight questions. Exactly nine. All ten. Add these chances up. You’ll get to a number slightly below 0.377. That is, Mary Lou would have just under a 37.7 percent chance of passing. The answer’s right and it’s easy to understand how it’s right. The only drawback is it’s a lot of calculating to get there.
So here’s the conceptually harder but faster way. It works because the problem says Mary Lou is as likely to get a problem wrong as right. So she’s as likely to get exactly ten questions right as exactly ten wrong. And as likely to get at least nine questions right as at least nine wrong. To get at least eight questions right as at least eight wrong. You see where this is going: she’s as likely to get at least six right as to get at least six wrong.
There’s exactly three possibilities for a ten-question assignment like this. She can get four or fewer questions right (six or more wrong). She can get exactly five questions right. She can get six or more questions right. The chance of the first case and the chance of the last have to be the same.
So, take 1 — the chance that one of the three possibilities will happen — and subtract the chance she gets exactly five problems right, which is a touch over 24.6 percent. So there’s just under a 75.4 percent chance she does not get exactly five questions right. It’s equally likely to be four or fewer, or six or more. Just-under-75.4 divided by two is just under 37.7 percent, which is the chance she’ll pass as the problem’s given. It’s trickier to see why that’s right, but it’s a lot less calculating to do. That’s a common trade-off.
Ruben Bolling’s Super-Fun-Pax Comix rerun for the 23rd is an aptly titled installment of A Million Monkeys At A Million Typewriters. It reminds me that I don’t remember if I’d retired the monkeys-at-typewriters motif from Reading the Comics collections. If I haven’t I probably should, at least after making a proper essay explaining what the monkeys-at-typewriters thing is all about.
Ted Shearer’s Quincy from the 28th of February, 1978 reveals to me that pocket calculators were a thing much earlier than I realized. Well, I was too young to be allowed near stuff like that in 1978. I don’t think my parents got their first credit-card-sized, solar-powered calculator that kind of worked for another couple years after that. Kids, ask about them. They looked like good ideas, but you could use them for maybe five minutes before the things came apart. Your cell phone is so much better.
Bil Watterson’s Calvin and Hobbes rerun for the 24th can be classed as a resisting-the-word-problem joke. It’s so not about that, but who am I to slow you down from reading a Calvin and Hobbes story?
Garry Trudeau’s Doonesbury rerun for the 24th started a story about high school kids and their bad geography skills. I rate it as qualifying for inclusion here because it’s a mathematics teacher deciding to include more geography in his course. I was amused by the week’s jokes anyway. There’s no hint given what mathematics Gil teaches, but given the links between geometry, navigation, and geography there is surely something that could be relevant. It might not help with geographic points like which states are in New England and where they are, though.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 24th is built on a plot point from Carl Sagan’s science fiction novel Contact. In it, a particular “message” is found in the digits of π. (By “message” I mean a string of digits that are interesting to us. I’m not sure that you can properly call something a message if it hasn’t got any sender and if there’s not obviously some intended receiver.) In the book this is an astounding thing because the message can’t be; any reasonable explanation for how it should be there is impossible. But short “messages” are going to turn up in π also, as per the comic strips.
I assume the peer review would correct the cartoon mathematicians’ unfortunate spelling of understanding.
It turns out last Saturday only had the one comic strip that was even remotely on point for me. And it wasn’t very on point either, but since it’s one of the Creators.com strips I’ve got the strip to show. That’s enough for me.
Ruben Bolling’s Super-Fun-Pak Comix for the 8th is an installation of They Came From The Third Dimension. “Dimension” is one of those oft-used words that’s come loose of any technical definition. We use it in mathematics all the time, at least once we get into Introduction to Linear Algebra. That’s the course that talks about how blocks of space can be stretched and squashed and twisted into each other. You’d expect this to be a warmup act to geometry, and I guess it’s relevant. But where it really pays off is in studying differential equations and how systems of stuff changes over time. When you get introduced to dimensions in linear algebra they describe degrees of freedom, or how much information you need about a problem to pin down exactly one solution.
It does give mathematicians cause to talk about “dimensions of space”, though, and these are intuitively at least like the two- and three-dimensional spaces that, you know, stuff moves in. That there could be more dimensions of space, ordinarily inaccessible, is an old enough idea we don’t really notice it. Perhaps it’s hidden somewhere too.
Amanda El-Dweek’s Amanda the Great of the 9th started a story with the adult Becky needing to take a mathematics qualification exam. It seems to be prerequisite to enrolling in some new classes. It’s a typical set of mathematics anxiety jokes in the service of a story comic. One might tsk Becky for going through university without ever having a proper mathematics class, but then, I got through university without ever taking a philosophy class that really challenged me. Not that I didn’t take the classes seriously, but that I took stuff like Intro to Logic that I was already conversant in. We all cut corners. It’s a shame not to use chances like that, but there’s always so much to do.
Mark Anderson’s Andertoons for the 10th relieves the worry that Mark Anderson’s Andertoons might not have got in an appearance this week. It’s your common kid at the chalkboard sort of problem, this one a kid with no idea where to put the decimal. As always happens I’m sympathetic. The rules about where to move decimals in this kind of multiplication come out really weird if the last digit, or worse, digits in the product are zeroes.
Mel Henze’s Gentle Creatures is in reruns. The strip from the 10th is part of a story I’m so sure I’ve featured here before that I’m not even going to look up when it aired. But it uses your standard story problem to stand in for science-fiction gadget mathematics calculation.
Dave Blazek’s Loose Parts for the 12th is the natural extension of sleep numbers. Yes, I’m relieved to see Dave Blazek’s Loose Parts around here again too. Feels weird when it’s not.
John Deering’s Strange Brew for the 13th is a “math club” joke featuring horses. Oh, it’s a big silly one, but who doesn’t like those too?
Dan Thompson’s Brevity for the 14th is one of the small set of punning jokes you can make using mathematician names. Good for the wall of a mathematics teacher’s classroom.
Shaenon K Garrity and Jefferey C Wells’s Skin Horse for the 14th is set inside a virtual reality game. (This is why there’s talk about duplicating objects.) Within the game, the characters are playing that game where you start with a set number (in this case 20) tokens and take turn removing a couple of them. The “rigged” part of it is that the house can, by perfect play, force a win every time. It’s a bit of game theory that creeps into recreational mathematics books and that I imagine is imprinted in the minds of people who grow up to design games.
I admit I’ve been a little unnerved lately. Between the A To Z project and the flood of mathematics-themed jokes from Comic Strip Master Command — and miscellaneous follies like my WordPress statistics-reading issues — I’ve had a post a day for several weeks now. The streak has to end sometime, surely, right? So it must, but not today. I admit the bunch of comics mentioning mathematical topics the past couple days was more one of continuing well-explored jokes rather than breaking new territory. But every comic strip is somebody’s first, isn’t it? (That’s an intimidating thought.)
Disney’s Mickey Mouse (June 6, rerun from who knows when) is another example of the word problem that even adults can’t do. I think it’s an interesting one for being also a tongue-twister. I tend to think of this sort of problem as a calculus question, but that’s surely just that I spend more time with calculus than with algebra or simpler arithmetic.
Justin Thompson’s Mythtickle (June 6, again a rerun) is about the curious way that objects are mostly empty space. The first panel shows on the alien’s chalkboard legitimate equations from quantum mechanics. The first line describes (in part) a function called psi that describes where a particle is likely to be found over time. The second and third lines describe how the probability distribution — where a particle is likely to be found — changes over time.
Doug Bratton’s Pop Culture Shock Therapy (July 7) just name-drops mathematics as something a kid will do badly in. In this case the kid is Calvin, from Calvin and Hobbes. While it’s true he did badly in mathematics I suspect that’s because it’s so easy to fit an elementary-school arithmetic question and a wrong answer in a single panel.
I’m sorry to have fallen behind on my mathematics-comics posts, but I’ve been very busy wielding a cudgel at Microsoft IIS all week in the service of my day job. And since I telecommute it’s quite hard to convincingly threaten the server, however much it deserves it. Sorry. Comic Strip Master Command decided to send me three hundred billion gazillion strips, too, so this is going to be a bit of a long post.
Jenny Campbell’s Flo and Friends (January 19) is almost a perfect example of the use of calculus as a signifier of “something really intelligent people think of”. Which is flattening to mathematicians, certainly, although I worry that attitude does make people freeze up in panic when they hear that they have to take calculus.
The Amazing Yet Tautological feature of Ruben Bolling’s Super-Fun-Pak Comix (January 19) lives up to its title, at least provided we are all in agreement about what “average” means. From context this seems to be the arithmetic mean — that’s usually what people, mathematicians included mean by “average” if they don’t specify otherwise — although you can produce logical mischief by slipping in an alternate average, such as the “median” — the amount that half the results are less than and half are greater than — or the “mode” — the most common result. There are other averages too, but they’re not so often useful. On the 21st Super-Fun-Pak Comix returned with another installation of Chaos Butterfly, by the way.
One of the little challenges in writing about mathematics-themed comics is one of pacing: how often should I do a roundup? Posting weekly, say, helps figure out a reasonable posting schedule for those rare moments when I’m working ahead of deadline, but that leaves the problem of weeks that just don’t have anything. Waiting for a certain number of comics before writing about them seems more reasonable, but then I have to figure how many comics are enough. I’ve settled into five-to-six as my threshold for a new post, but that can mean I have weeks where it seems like I’m doing nothing but comic strips posts. And then there’s conditions like this one where Comic Strip Master Command had its cartoonists put up just enough that I’d started composing a fresh post, and then tossed in a whole bunch more the next day. It’s like they’re trying to shake me by having too many strips to write about. I’d have though they’d be flattered to have me writing about them so.
Marc Anderson’s Andertoons (September 12) belongs to that vein of humor about using technology words to explain stuff to kids. I admit I’m vague enough on the concept of mashups that I can accept that it might be a way of explaining addition, but it feels like it might also be a way of describing multiplication or for that matter the composition of functions. I suppose the kids would be drawn as older in those cases, though.
Bill Amend’s FoxTrot (September 13, rerun) does a word problem joke, but it does have the nice beat in the penultimate panel of Paige running a sanity check and telling at a glance that “two dollars” can’t possibly be the right answer. Sanity checks are nice things to have; they don’t guarantee against making mistakes, but they at least provide some protection against the easiest mistakes, and having some idea of what an answer could plausibly be might help in working out the answer. For example, if Paige had absolutely no idea how to set up equations for this problem, she could reason that the apple and the orange have to cost something from 1 to 29 cents, and could try out prices until finding something that satisfies both requirements. This is an exhausting method, but it would eventually work, too, and sometimes “working eventually” is better than “working cleverly”.
Bill Schorr’s The Grizzwells (September 13) starts out by playing on the fact that “yard” has multiple meanings; it also circles around one of those things that distinguishes word problems from normal mathematics. A word problem, by convention, normally contains exactly the information needed to solve what’s being asked — there’s neither useless information included nor necessary information omitted, except if the question-writer has made a mistake. In a real world application, figuring out what you need, and what you don’t need, is part of the work, possibly the most important part of the work. So to answer how many feet are in a yard, Gunther (the bear) is right to ask more questions about how big the yard is, as a start.
Steve Kelley and Jeff Parker’s Dustin (September 14) is about one of the applications for mental arithmetic that people find awfully practical: counting the number of food calories that you eat. Ed’s point about it being convenient to have food servings be nice round numbers, as they’re easier to work with, is a pretty good one, and it’s already kind of accounted for in food labelling: it’s permitted (in the United States) to round off calorie counts to the nearest ten or so, on the rather sure grounds that if you are counting calories you’d rather add 70 to the daily total than 68 or 73. Don’t read the comments thread, which includes the usual whining about the Common Core and the wild idea that mental arithmetic might be well done by working out a calculation that’s close to the one you want but easier to do and then refining it to get the accuracy you need.
Mac and Bill King’s Magic In A Minute kids activity panel (September 14) presents a magic trick that depends on a bit of mental arithmetic. It’s a nice stunt, although it is certainly going to require kids to practice things because, besides dividing numbers by 4, it also requires adding 6, and that’s an annoying number to deal with. There’s also a nice little high school algebra problem to be done in explaining why the trick works.
Bill Watterson’s Calvin and Hobbes (September 15, rerun) includes one of Hobbes’s brilliant explanations of how arithmetic works, and if I haven’t wasted the time spent memorizing the strips where Calvin tries to do arithmetic homework then Hobbes follows up tomorrow with imaginary numbers. Can’t wait.
Jef Mallet’s Frazz (September 15) expresses skepticism about a projection being made for the year 2040. Extrapolations and interpolations are a big part of numerical mathematics and there’s fair grounds to be skeptical: even having a model of whatever your phenomenon is that accurately matches past data isn’t a guarantee that there isn’t some important factor that’s been trivial so far but will become important and will make the reality very different from the calculations. But that hardly makes extrapolations useless: for one, the fact that there might be something unknown which becomes important is hardly a guarantee that there is. If the modelling is good and the reasoning sound, what else are you supposed to use for a plan? And of course you should watch for evidence that the model and the reality aren’t too very different as time goes on.