I have not kept secret that I’ve had little energy lately. I hope that’s changing but can do little more than hope. I find it strange that my lack of energy seems to be matched by Comic Strip Master Command. Last week saw pretty slim pickings for mathematically-themed comics. Here’s what seems worth the sharing from my reading.
Lincoln Peirce’s Big Nate for the 22nd is a Pi Day joke, displaced to the prank day at the end of Nate’s school year. It’s also got a surprising number of people in the comments complaining that 3.1416 is only an approximation to π. It is, certainly, but so is any representation besides π or a similar mathematical expression. And introducing it with 3.1416 gives the reader the hint that this is about a mathematics expression and not an arbitrary symbol. It’s important to the joke that this be communicated clearly, and it’s hard to think of better ways to do that.
Dave Whamond’s Reality Check for the 24th is another in the line of “why teach algebra instead of something useful” strips. There are several responses. One is that certainly one should learn how to do a household budget; this was, at least back in the day, called home economics, and it was a pretty clear use of mathematics. Another is that a good education is about becoming literate in all the great thinking of humanity: you should come out knowing at least something coherent about mathematics and literature and exercise and biology and music and visual arts and more. Schools often fail to do all of this — how could they not? — but that’s not reason to fault them on parts of the education that they do. And anther is that algebra is about getting comfortable working with numbers before you know just what they are. That is, how to work out ways to describe a thing you want to know, and then to find what number (or range of numbers) that is. Still, these responses hardly matter. Mathematics has always lived in a twin space, of being both very practical and very abstract. People have always and will always complain that students don’t learn how to do the practical well enough. There’s not much changing that.
Charles Schulz’s Peanuts Begins for the 26th sees Violet challenge Charlie Brown to say what a non-perfect circle would be. I suppose this makes the comic more suitable for a philosophy of language blog, but I don’t know any. To be a circle requires meeting a particular definition. None of the things we ever point to and call circles meets that. We don’t generally have trouble connecting our imperfect representations of circles to the “perfect” ideal, though. And Charlie Brown said something meaningful in describing his drawing as being “a perfect circle”. It’s tricky pinning down exactly what it is, though.
All right, I can be a little more clear. By the inverse I mean subtraction is the name the name we give to adding the additive inverse of something. It’s what lets addition be a group action. That is, we write to mean we find whatever number, added to b, gives us 0. Then we add that to a. We do this pretty often, so it’s convenient to have a name for it. The word “subtraction” appears in English from about 1400. It grew from the Latin for “taking away”. By about 1425 the word has its mathematical meaning. I imagine this wasn’t too radical a linguistic evolution.
All right, so some other thoughts. What’s so interesting about subtraction that it’s worth a name? We don’t have a particular word for reversing, say, a permutation. But don’t go very far in school not thinking about inverting an addition. Must come down to subtraction’s practical use in finding differences between things. Often in figuring out change. Debts at least. Nobody needs the inverse of a permutation unless they’re putting a deck of cards back in order.
Subtraction has other roles, though. Not so much in mathematics, but in teaching us how to learn about mathematics. For example, subtraction gives us a good reason to notice zero. Zero, the additive identity, is implicit to addition. But if you’re learning addition, and you think of it as “put these two piles of things together into one larger pile”? What good does an empty pile do you there? It’s easy to not notice there’s a concept there. But subtraction, taking stuff away from a pile? You can imagine taking everything away, and wanting a word for that. This isn’t the only way to notice zero is worth some attention. It’s a good way, though.
There’s more, though. Learning subtraction teaches us limits of what we can do, mathematically. We can add 3 to 7 or, if it’s more convenient, 7 to 3. But we learn from the start that while we can subtract 3 from 7, there’s no subtracting 7 from 3. This is true when we’re learning arithmetic and numbers are all positive. Some time later we ask, what happens if we go ahead and do this anyway? And figure out a number that makes sense as the answer to “what do you get subtracting 7 from 3”? This introduces us to the negative numbers. It’s a richer idea of what it is to have numbers. We can start to see addition and subtraction as expressions of the same operation.
But we also notice they’re not quite the same. As mentioned, addition can be done in any order. If I need to do 7 + 4 + 3 + 6 I can decide I’d rather do 4 + 6 + 7 + 3 and make that 10 + 10 before getting to 20. This all simplifies my calculating. If I need to do 7 – 4 – 3 – 6 I get into a lot of trouble if I simplify my work by writing 4 – 6 – 7 – 3 instead. Even if I decide I’d rather take the 3 – 6 and turn that into a negative 3 first, I’ve made a mess of things.
The first property this teaches us to notice we call “commutativity”. Most mathematical operations don’t have that. But a lot of the ones we find useful do. The second property this points out is “associativity”, which more of the operations we find useful have. It’s not essential that someone learning how to calculate know this is a way to categorize mathematics operations. (I’ve read that before the New Math educational reforms of the 1960s, American elementary school mathematics textbooks never mentioned commutativity or associativity.) But I suspect it is essential that someone learning mathematics learn the things you can do come in families.
So let me mention division, the inverse of multiplication. (And that my chosen theme won’t let me get to in sequence.) Like subtraction, division refuses to be commutative or associative. Subtraction prompts us to treat the negative numbers as something useful. In parallel, division prompts us to accept fractions as numbers. (We accepted fractions as numbers long before we accepted negative numbers, mind. Anyone with a pie and three friends has an interest in “one-quarter” that they may not have with “negative four”.) When we start learning about numbers raised to powers, or exponentials, we have questions ready to ask. How do the operations behave? Do they encourage us to find other kinds of number?
And we also think of how to patch up subtraction’s problems. If we want subtraction to be a kind of addition, we have to get precise about what that little subtraction sign means. What we’ve settled on is that is shorthand for , where is the additive inverse of .
Once we do that all subtraction’s problems with commutativity and associativity go away. 7 – 4 – 3 – 6 becomes 7 + (-4) + (-3) + (-6), and that we can shuffle around however convenient. Say, to 7 + (-3) + (-4) + (-6), then to 7 + (-3) + (-10), then to 4 + (-10), and so -6. Thus do we domesticate a useful, wild operation like subtraction.
Any individual subtraction has one right answer. There are many ways to get there, though. I had learned, for example, to do a problem such as 738 minus 451 by subtracting one column of numbers at a time. Right to left, so, subtracting 8 minus 1, and then 3 minus 5, and after the borrowing then 6 minus 4. I remember several elementary school textbooks explaining borrowing as unwrapping rolls of dimes. It was a model well-suited to me.
We don’t need to, though. We can go from the left to the right, doing 7 minus 4 first and 8 minus 1 last. We can go through and figure out all the possible carries before doing any work. There’s a slick method called partial differences which skips all the carrying. But it demands writing out several more intermediate terms. This uses more paper, but if there isn’t a paper shortage, so what?
There are more ways to calculate. If we turn things over to a computer, we’re likely to do subtraction using a complements technique. When I say computer you likely think electronic computer, or did right up to the adjective there. But mechanical computers were a thing too. Blaise Pascal’s computing device of the 1650s used nines’ complements to subtract on the gears that did addition. Explaining the trick would take me farther afield than I want to go now. But, you know how, like, 6 plus 3 is 9? So you can turn a subtraction of 6 into an addition of 3. Or a subtraction of 3 into an addition of 6. Plus some bookkeeping.
A digital computer is likely to use ones’ complements, representing every number as a string of 0’s and 1’s. This has great speed advantages. The complement of 0 is 1 and vice-versa, and it’s very quick for a computer to swap between 0 and 1. Subtraction by complements is different and, to my eye, takes more steps. But they might be steps you do better.
One more thought subtraction gives us, though. In a previous paragraph I wrote out 7 – 4, and also wrote 7 + (-4). We use the symbol – for two things. Do those two uses of – mean the same thing? You may think I’m being fussy here. After all, the value of -4 is the same as the value of 0 – 4. And even a fussy mathematician says whichever of “minus four” and “negative four” better fits the meter of the sentence. But our friends in the philosophy department would agree this is a fair question. Are we collapsing two related ideas together by using the same symbol for them?
My inclination is to say that the – of -4 is different from the – in 0 – 4, though. The – in -4 is a unary operation: it means “give me the inverse of the number on the right”. The – in 0 – 4 is a binary operation: it means “subtract the number on the right from the number on the left”. So I would say these are different things sharing a symbol. Unfortunately our friends in the philosophy department can’t answer the question for us. The university laid them off four years ago, part of society’s realignment away from questions like “how can we recognize when a thing is true?” and towards “how can we teach proto-laborers to use Excel macros?”. We have to use subtraction to expand our thinking on our own.
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.
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.
And here’s the rest of last week’s mathematically-themed comic strips. On reflection, none of them are so substantially about the mathematics they mention for me to go into detail. Again, Comic Strip Master Command is helping me rebuild my energies after the A-to-Z wrapped up. I appreciate it, folks, but would like, you know, two or three strips a week I can sink my teeth into.
Charles Schulz’s Peanuts rerun for the 11th sees Sally Brown working out metric system unit conversions. The strip originally ran the 13th of December, 1972, a year when people in the United States briefly thought there might ever be a reason to use the prefix “deci-” for something besides decibels. “centi-” for anything besides “centimeter” is pretty dodgy too.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 13th is titled “Do Not Date A Mathematician”. This seems personal. The point here is the mathematician believing her fiancee has “demonstrated a poor understanding of probability” by declaring his belief in soulmates. The joke seems to be missing some key points, though. Just declaring a belief in soulmates doesn’t say anything about his understanding of probability. If we suppose that he believed every person had exactly one soulmate, and that these soulmates were uniformly distributed across the world’s population, and that people routinely found their soulmates. But if those assumptions aren’t made then you can’t say that the fiancee is necessarily believing in something improbable.
Chris Browne’s Raising Duncan rerun for the 23rd has a man admitting bad mathematics skills for why he can’t count the ways he loves his wife. The strip originally ran the 27th of September, 2003. (The strip was short-lived, and is in perpetual reruns. It may be worth reading at least one time through, though, since the pairs of main characters in it are eagerly in love, without being sappy about it, and it’s pleasant seeing people enthusiastic about each other. This is the strip that had the exchange “Marry me!” “I did!” “Marry me more!” “Okay!” that keeps bringing me cheer and relationship goals.)
John Hambrock’s The Brilliant Mind of Edison Lee for the 1st of October is a calendar joke. Well, many of the months used to have names that denoted their count. Month names have changed more than you’d think. For a while there every Roman Emperor was renaming months after himself. Most of these name changes did not stick. Lucius Aurelius Commodus, who reined from 177 to 192, gave all twelve months one or another of his names.
The past week included another Friday the 13th. Several comic strips found that worth mention. So that gives me a theme by which to name this look over the comic strips.
Charles Schulz’s Peanuts rerun for the 12th presents a pretty wordy algebra problem. And Peppermint Patty, in the grips of a math anxiety, freezing up and shutting down. One feels for her. Great long strings of words frighten anyone. The problem seems a bit complicated for kids Peppermint Patty’s and Franklin’s age. But the problem isn’t helping. One might notice, say, that a parent’s age will be some nice multiple of a child’s in a year or two. That in ten years a man’s age will be 14 greater than the combined age of their ages then? What imagination does that inspire?
Grant Peppermint Patty her fears. The situation isn’t hopeless. It helps to write out just what know, and what we would like to know. At least what we would like to know if we’ve granted the problem worth solving. What we would like is to know the man’s age. That’s some number; let’s call it M. What we know are things about how M relates to his daughter’s and his son’s age, and how those relate to one another. Since we know several things about the daughter’s age and the son’s age it’s worth giving those names too. Let’s say D for the daughter’s age and S for the son’s.
So. We know the son is three years older than the daughter. This we can write as . We know that in one year, the man will be six times as old as the daughter is now. In one year the man will be M + 1 years old. The daughter’s age now is D; six times that is 6D. So we know that . In ten years the man’s age will be M + 10; the daughter’s age, D + 10; the son’s age, S + 10. In ten years, M + 10 will be 14 plus D + 10 plus S + 10. That is, . Or if you prefer, . Or even, .
So this is a system of three equation, all linear, in three variables. This is hopeful. We can hope there will be a solution. And there is. There are different ways to find an answer. Since I’m grading this, you can use the one that feels most comfortable to you. The problem still seems a bit advanced for Peppermint Patty and Franklin.
Julie Larson’s The Dinette Set rerun for the 13th has a bit of talk about a mathematical discovery. The comic is accurate enough for its publication. In 2008 a number known as M43112609 was proven to be prime. The number, 243,112,609 – 1, is some 12,978,189 digits long. It’s still the fifth-largest known prime number (as I write this).
Prime numbers of the form 2N – 1 for some whole number N are known as Mersenne primes. These are named for Marin Mersenne, a 16th century French friar and mathematician. They’re a neat set of numbers. Each Mersenne prime matches some perfect number. Nobody knows whether there are finite or infinitely many Mersenne primes. Every even perfect number has a form that matches to some Mersenne prime. It’s unknown whether there are any odd perfect numbers. As often happens with number theory, the questions are easy to ask but hard to answer. But all the largest known prime numbers are Mersenne primes; they’re of a structure we can test pretty well. At least that electronic computers can test well; the last time the largest known prime was found by mere mechanical computer was 1951. The last time a non-Mersenne was the largest known prime was from 1989 to 1992, and before that, 1951.
T Shepherd’s Snow Sez for the 13th finishes off the unlucky-13 jokes. It observes that whatever a symbol might connote generally, your individual circumstances are more important. There are people for whom 13 is a good omen, or for whom Mondays are magnificent days, or for whom black cats are lucky.
These are all the comics I can write paragraphs about. There were more comics mentioning mathematics last week. Here were some of them:
I concede I am late in wrapping up last week’s mathematically-themed comics. But please understand there were important reasons for my not having posted this earlier, like, I didn’t get it written in time. I hope you understand and agree with me about this.
Bill Griffith’s Zippy the Pinhead for the 9th brings up mathematics in a discussion about perfection. The debate of perfection versus “messiness” begs some important questions. What I’m marginally competent to discuss is the idea of mathematics as this perfect thing. Mathematics seems to have many traits that are easy to think of as perfect. That everything in it should follow from clearly stated axioms, precise definitions, and deductive logic, for example. This makes mathematics seem orderly and universal and fair in a way that the real world never is. If we allow that this is a kind of perfection then … does mathematics reach it?
Even the idea of a “precise definition” is perilous. If it weren’t there wouldn’t be so many pop mathematics articles about why 1 isn’t a prime number. It’s difficult to prove that any particular set of axioms that give us interesting results are also logically consistent. If they’re not consistent, then we can prove absolutely anything, including that the axioms are false. That seems imperfect. And few mathematicians even prepare fully complete, step-by-step proofs of anything. It takes ridiculously long to get anything done if you try. The proofs we present tend to show, instead, the reasoning in enough detail that we’re confident we could fill in the omitted parts if we really needed them for some reason. And that’s fine, nearly all the time, but it does leave the potential for mistakes present.
Zippy offers up a perfect parallelogram. Making it geometry is of good symbolic importance. Everyone knows geometric figures, and definitions of some basic ideas like a line or a circle or, maybe, a parallelogram. Nobody’s ever seen one, though. There’s never been a straight line, much less two parallel lines, and even less the pair of parallel lines we’d need for a parallellogram. There can be renderings good enough to fool the eye. But none of the lines are completely straight, not if we examine closely enough. None of the pairs of lines are truly parallel, not if we extend them far enough. The figure isn’t even two-dimensional, not if it’s rendered in three-dimensional things like atoms or waves of light or such. We know things about parallelograms, which don’t exist. They tell us some things about their shadows in the real world, at least.
Mark Litzler’s Joe Vanilla for the 9th is a play on the old joke about “a billion dollars here, a billion dollars there, soon you’re talking about real money”. As we hear more about larger numbers they seem familiar and accessible to us, to the point that they stop seeming so big. A trillion is still a massive number, at least for most purposes. If you aren’t doing combinatorics, anyway; just yesterday I was doing a little toy problem and realized it implied 470,184,984,576 configurations. Which still falls short of a trillion, but had I made one arbitrary choice differently I could’ve blasted well past a trillion.
Ruben Bolling’s Super-Fun-Pak Comix for the 9th is another monkeys-at-typewriters joke, that great thought experiment about probability and infinity. I should add it to my essay about the Infinite Monkey Theorem. Part of the joke is that the monkey is thinking about the content of the writing. This doesn’t destroy the prospect that a monkey given enough time would write any of the works of William Shakespeare. It makes the simple estimates of how unlikely that is, and how long it would take to do, invalid. But the event might yet happen. Suppose this monkey decided there was no credible way to delay Hamlet’s revenge to Act V, and tried to write accordingly. Mightn’t the monkey make a mistake? It’s easy to type a letter you don’t mean to. Or a word you don’t mean to. Why not a sentence you don’t mean to? Why not a whole act you don’t mean to? Impossible? No, just improbable. And the monkeys have enough time to let the improbable happen.
Eric the Circle for the 10th, this one by Kingsnake, declares itself set in “the 20th dimension, where shape has no meaning”. This plays on a pop-cultural idea of dimensions as a kind of fairyland, subject to strange and alternate rules. A mathematician wouldn’t think of dimensions that way. 20-dimensional spaces — and even higher-dimensional spaces — follow rules just as two- and three-dimensional spaces do. They’re harder to draw, certainly, and mathematicians are not selected for — or trained in — drawing, at least not in United States schools. So attempts at rendering a high-dimensional space tend to be sort of weird blobby lumps, maybe with a label “N-dimensional”.
And a projection of a high-dimensional shape into lower dimensions will be weird. I used to have around here a web site with a rotatable tesseract, which would draw a flat-screen rendition of what its projection in three-dimensional space would be. But I can’t find it now and probably it ran as a Java applet that you just can’t get to work anymore. Anyway, non-interactive videos of this sort of thing are common enough; here’s one that goes through some of the dimensions of a tesseract, one at a time. It’ll give some idea how something that “should” just be a set of cubes will not look so much like that.
Steve Kelly and Jeff Parker’s Dustin for the 11th is a variation on the “why do I have to learn this” protest. This one is about long division and the question of why one needs to know it when there’s cheap, easily-available tools that do the job better. It’s a fair question and Hayden’s answer is a hard one to refute. I think arithmetic’s worth knowing how to do, but I’ll also admit, if I need to divide something by 23 I’m probably letting the computer do it.
One is that, as we’ve thought of counting numbers, there is always “one more”. This doesn’t have to be. We could work with perfectly good number systems that have a largest number. We do, in fact. Every computer programming language has some largest integer that it will deal with. If you need a larger number, you have to do something clever. Your clever idea will let you address some range of bigger numbers, but it too will have a maximum. We’ve set those limits large enough that, usually, they’re not an inconvenience. They’re still there.
But those limits are forced on us by the many failings of matter. What when we get just past Plato’s line’s division, into the reasoning of pure mathematics? There we can set up counting numbers. The standard way to do this is to suppose there is a number “1”. And to suppose that, for any counting number we have, there is a successor, a number one-plus-that. If Joey were to ask why there has to be, all Dennis could do is shrug. This makes an axiom out of there always being one more. If you don’t like it, make some other arithmetic. Anyway we only understand any of this using fallible matter, so good luck.
This progression can be heady, though. The counting numbers are probably the most understandable infinitely large set there is. Thinking about them seriously can induce the sort of dizzy awe that pondering Deep Time or the vastness of space can do. That seems a bit above Dennis’s age level, but some people are stricken with the infinite sooner than others are.
Charles Schulz’s Peanuts Begins rerun for the 2nd has Charlie Brown dismiss arithmetic as impractical. It fits the motif of mathematics as an unworldly subject. There’s the common joke that pure mathematics even dreams of being of no use to anyone. Arithmetic, though, has always been a practical subject. It introduces us to many abstract ideas, particularly group theory. This subject looks at what we can do with systems that work like arithmetic without necessarily having numbers, or anything that works with numbers.
And a couple of comic strips mentioned mathematics, although in too slight a way to discuss. Dana Simpson’s Phoebe and her Unicorn on the 30th of April started a sequence in which doodles on Phoebe’s homework came to life. That it’s mathematics homework was mostly incidental. I’m open to the argument that mathematics encourages doodling in a way that, say, spelling does not. I’d also be open to the argument you aren’t doing geometry if you don’t doodle. Anyway. Dan Thompson’s Brevity for the 2nd of May features Sesame Street’s Count von Count. It’s a bit of wordplay on the use of “numbers” for songs. And, of course, the folkloric tradition of vampires as compulsive counters.
There were just enough mathematically-themed comic strips last week to make two editions for this coming week. All going well I’ll run the other half on either Wednesday or Thursday. There is a point that isn’t quite well, which is that one of the comics is in dubious taste. I’ll put that at the end, behind a more specific content warning. In the meanwhile, you can find this and hundreds of other Reading the Comics posts at this link.
Thaves’s Frank and Ernest for the 11th is wordplay, built on the conflation of “negative” as in numbers and “negative” as in bad. I’m not sure the two meanings are unrelated. The word ‘negative’ itself derives from the Latin word meaning to deny, which sounds bad. It’s easy to see why the term would attach to what we call negative numbers. A number plus its negation leaves us zero, a nothing. But it does make the negative numbers sound like bad things to have around, or to have to deal with. The convention that a negative number is less than zero implies that the default choice for a number is one greater than zero. And the default choice is usually seen as the good one, with everything else a falling-away. Still, -7 is as legitimate a number as 7 is; it’s we who think one is better than another.
J C Duffy’s Lug Nuts for the 11th has the Dadaist panel present prime numbers as a way to communicate. I suspect Duffy’s drawing from speculations about how to contact alien intelligences. One problem with communicating with the truly alien is how to recognize there is a message being sent. A message too regular will look like a natural process, one conveying no more intelligence than the brightness which comes to most places at dawn and darkness coming at sunset. A message too information-packed, peculiarly, looks like random noise. We need an intermediate level. A signal that it’s easy to receive, and that is too hard to produce by natural processes.
Prime numbers seem like a good compromise. An intelligence that understands arithmetic will surely notice prime numbers, or at least work out quickly what’s novel about this set of numbers once given them. And it’s hard to imagine an intelligence capable of sending or receiving interplanetary signals that doesn’t understand arithmetic. (Admitting that yes, we might be ruling out conversational partners by doing this.) We can imagine a natural process that sends out (say) three pulses and then rests, or five pulses and rests. Or even draws out longer cycles: two pulses and a rest, three pulses and a rest five pulses and a rest, and then a big rest before restarting the cycle. But the longer the string of prime numbers, the harder it is to figure a natural process that happens to hit them and not other numbers.
We think, anyway. Until we contact aliens we won’t really know what it’s likely alien contact would be like. Prime numbers seem good to us, but — even if we stick to numbers — there’s no reason triangular numbers, square numbers, or perfect numbers might not be as good. (Well, maybe not perfect numbers; there aren’t many of them, and they grow very large very fast.) But we have to look for something particular, and this seems like a plausible particularity.
Charles Schulz’s Peanuts Begins for the 11th is an early strip, from the days when Lucy would look to Charlie Brown for information. And it’s a joke built on conflating ‘zero’ with ‘nothing’. Lucy’s right that zero times zero has to be something. That’s how multiplication works. That the number zero is something? That’s a tricky concept. I think being mathematically adept can blind one to how weird that is. If you’re used to how zero is the amount of a thing you have to have nothing of that thing, then we start to see what’s weird about it.
But I’m not sure the strip quite sets that up well. I think if Charlie Brown had answered that zero times zero was “nothing” it would have been right (or right enough) and Lucy’s exasperation would have flowed more naturally. As it is? She must know that zero is “nothing”; but then why would she figure “nothing times nothing” has to be something? Maybe not; it would have left Charlie Brown less reason to feel exasperated or for the reader to feel on Charlie Brown’s side. Young Lucy’s leap to “three” needs to be at least a bit illogical to make any sense.
Now to the last strip and the one I wanted to warn about. It alludes to gun violence and school shootings. If you don’t want to deal with that, you’re right. There’s other comic strips to read out there. And this for a comic that ran on the centennial of Armistice Day, which has to just be an oversight in scheduling the (non-plot-dependent) comic.
Four more comics from last week struck me as worth mentioning. Two of them are over sixty years old.
Incidentally, Walt Kelly’s Pogo first appeared in the newspaper seventy years ago today. I don’t know anyone rerunning the comics the way Skippy or Thimble Theatre (Popeye) or the like are, which is a shame. (Few if any strips would be on-point around here, but it’s still worth reading.) But I did think some of the folks around here would like to know.
Percy Crosby’s Skippy for the 25th is a vintage-1931 strip about the miseries of learning arithmetic. Skippy’s scheme to both improve by copying one another’s 50-percent-right papers is not necessarily a bad one. It depends on a couple things to work. For example, do they both get the same questions wrong? Possibly; it’d be natural for both students to do worse on the harder questions. But suppose that the questions Skippy and Sooky get wrong are independent of one another. That is, knowing that Skippy got a question right doesn’t affect our estimate of the probability whether Sooky got that question right. In that case, we’d expect both of them to get about 25% of the questions right. And at least one of them would get about 75% of the questions right. So, if they could copy the right answers, they could get a 25-point improvement. That’s pretty good.
Telling which are the right answers is hard. But, it’s typically easier to check whether an answer is right than it is to find an answer. Arithmetic is a point where this might not be usefully so. You can verify that 25 – 17 is indeed 8 by trying to calculate 17 + 8. But I don’t know that one equation is easier than the other.
Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde for the 26th is a percentages joke. Miss Latham is making the supposition that one hundred percent effort is needed to get the assignment done correctly. That’s fair if the full effort to make is “what effort it takes to do the assignment correctly”. Tautological, but indisputable. If the one-hundred-percent-effort is whatever’s considered the appropriate standard effort to make for an assignment this size … well, that’s harder to agree with. Some assignments, some days, are easy; some just aren’t. Depends on what’s being asked.
Bill Schorr’s The Grizzwells for the 27th says it’s about mathematics. The particular question is about how many quarts go into a gallon. Measurement questions like this do get bundled into mathematics. It’s a bit hard to say why, though. It’s arbitrary how big a unit is; all we really demand is that it be convenient for whatever we’re doing. It’s even more arbitrary what the subdivisions of a unit are. A quart — well, the name gives away, it should be a quarter of something bigger. But there’s no reason we couldn’t have divided a gallon into three pieces, or six, or twelve instead. We just didn’t happen to do that. And similarly for subdividing a quart (or whatever name it would get, if it were a sixth of a gallon).
I suppose it’s from thinking of arithmetic as a tool for clerks and shopkeepers. These calculations would need to carry along units. Even the currency might need to carry units. Decimal currency obscures the units. Older-style pound-shilling-pence units (or whatever they were called in the local language) don’t allow that. So I’m guessing that it was natural to think of, say, “quadruple three quarts” as the same sort of problem as “one-sixth of 8s/4d”.
Charles Schulz’s Peanuts Begins for the 29th speaks of “a perfect circle”. Violet asks an excellent question. But to say “a perfect circle” does communicate something. We name things like circles and lines and squares and agree they have certain properties. Also that the circles or lines or squares that we see in the world don’t have those properties. We might emphasize that something is a perfect circle or a straight line or something, to insist that it approaches this ideal of circle-ness. I’m not well-versed in the philosophy of mathematics. But it does seem hard to avoid Platonist thoughts about it. It’s hard to do geometry without pictures. But we insist to ourselves that the pictures may lie to us.
I apologize for not having a more robust introduction here. My week’s been chopped up by concern with the health of the older of our rabbits. Today’s proved to be less alarming than we had feared, but it’s still a lot to deal with. I appreciate your kind thoughts. Thank you.
Meanwhile the comics from last week have led me to discover something really weird going on with the Mutt and Jeff reruns.
Charles Schulz’s Peanuts Classics for the 6th has the not-quite-fully-formed Lucy trying to count the vast. She’d spend a while trying to count the stars and it never went well. It does inspire the question of how to count things when doing a simple tally is too complicated. There are many mathematical approaches. Most of them are some kind of sampling. Take a small enough part that you can tally it, and estimate the whole based on what your sample is. This can require ingenuity. For example, when estimating our goldfish population, it was impossible to get a good sample at one time. When tallying the number of visible stars in the sky, we have the problem that the Galaxy has a shape, and there are more stars in some directions than in others. This is why we need statisticians.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 6th looks initially like it’s meant for a philosophy blog’s Reading the Comics post. It’s often fruitful in the study of ethics to ponder doing something that is initially horrible, but would likely have good consequences. Or something initially good, but that has bad effects. These questions challenge our ideas about what it is to do good or bad things, and whether transient or permanent effects are more important, and whether it is better to be responsible for something (or to allow something) by action or inaction.
It comes to mathematics in the caption, though, and with an assist from the economics department. Utilitarianism seems to offer an answer to many ethical problems. It posits that we need to select a primary good of society, and then act so as to maximize that good. This does have an appeal, I suspect even to people who don’t thrill of the idea of finding the formula that describes society. After all, if we know the primary good of society, why should we settle for anything but the greatest value of that good? It might be difficult in practice, say, to discount the joy a musician would bring over her lifetime with her performances fairly against the misery created by making her practice the flute after school when she’d rather be playing. But we can imagine working with a rough approximation, at least. Then the skilled thinkers point out even worse problems and we see why utilitarianism didn’t settle all the big ethical questions, even in principle.
The mathematics, though. As Weinersmith’s caption puts it, we can phrase moral dilemmas as problems of maximizing evil. Typically we pose them as ones of maximizing good. Or at least of minimizing evil. But if we have the mechanism in place to find where evil is maximized, don’t we have the tools to find where good is? If we can find the set of social parameters x, y, and z which make E(x, y, z) as big as possible, can’t we find where -E(x, y, z) is as big, too? And isn’t that then where E(x, y, z) has to be smallest?
And, sure. As long as the maximum exists, or the minimum exists. Maybe we can tell whether or not there is one. But this is why when you look at the mathematics of finding maximums you realize you’re also doing minimums, or vice-versa. Pretty soon you either start referring to what you find as extremums. Or you stop worrying about the difference between a maximum and a minimum, at least unless you need to check just what you have found. Or unless someone who isn’t mathematically expert looks at you wondering if you know the difference between positive and negative numbers.
Bud Fisher’s Mutt and Jeff for the 7th has run here before. Except that was before they redid the lettering; it was a roast beef in earlier iterations. I was thinking to drop Mutt and Jeff from my Reading the Comics routine before all these mysteries in the lettering turned up. Anyway. The strip’s joke starts with a work-rate problems. Given how long some people take to do a thing, how long does it take a different number of people to do a thing? These are problems that demand paying attention to units, to the dimensions of a thing. That seems to be out of fashion these days, which is probably why these questions get to be baffling. But if eating a ham takes 25 person-minutes to do, and you have ten persons eating, you can see almost right away how long to expect it to take. If the ham’s the same size, anyway.
Olivia Jaimes’s Nancy for the 7th is built on a spot of recreational mathematics. Also on the frustration one can have when a problem looks like it’s harmless innocent fun and turns out to take just forever and you’re never sure you have the answers just right. The commenters on GoComics.com have settled on 18. I’m content with that answer.
There were a bunch of mathematically-themed comic strips this past week. A lot of them are ones I’d seen before. One of them is a bit risque and I’ve put that behind a cut. This saves me the effort of thinking up a good nonsense name to give this edition, so there’s that going for me too.
Bill Amend’s FoxTrot Classics for the 24th of May ought to have run last Sunday, but I wasn’t able to make time to write about it. It’s part of a sequence of Jason tutoring Paige in geometry. She’s struggling with the areas of common shapes which is relatable. Many of these area formulas could be kept straight by thinking back to rectangles. The size of the area is equal to the length of the base times the length of the height. From that you could probably reason right away the area of a trapezoid. It would have the same area as a rectangle with a base of length the mean length of the trapezoid’s different-length sides. The parallelogram works like the rectangle, length of the base times the length of the height. That you can convince yourself of by imagining the parallelogram. Then imagine slicing a right triangle off one of its sides. Move that around to the other side. Put it together right and you have a rectangle. Already know the area of a rectangle. The triangle, then, you can get by imagining two triangles of the same size and shape. Rotate one of the triangles 180 degrees. Slide it over, so the two triangles touch. Do this right and you have a parallelogram and so you know the area. The triangle’s half the area of that parallelogram.
The circle, I don’t know. I think just remember that if someone says “pi” they’re almost certainly going to follow it with either “r squared” or “day”. One of those suggests an area; the other doesn’t. Best I can do.
Allison Barrows’s PreTeena rerun for the 27th discusses self-esteem as though it were a good thing that children ought to have. This is part of the strip’s work to help build up the Old Person Complaining membership that every comics section community group relies on. But. There is mathematics in Jeri’s homework. Not mathematics in the sense of something particular to calculate. There’s just nothing to do there. But it is mathematics, and useful mathematics, to work out the logic of how to satisfy multiple requirements. Or, if it’s impossible to satisfy them all at once, then to come as near satisfying them as possible. These kinds of problems are considered optimization or logistics problems. Most interesting real-world examples are impossibly hard, or at least become impossibly hard before you realize it. You can make a career out of doing as best as possible in the circumstances.
Charles Schulz’s Peanuts rerun for the 27th features an extended discussion by Lucy about the nature of … well, she explicitly talks about “nothing”. Is she talking about zero? Probably; you have to get fairly into mathematics or philosophy to start worrying about the difference between the number zero and the idea of nothing. In Algebra, mathematicians learn to work with systems of things that work like numbers enough that you can add and subtract and multiply them together, without committing to the idea that they’re working with numbers. They will have something that works like zero, though, a “nothing” that can be added to or subtracted from anything without changing it. And for which multiplication turns something into that “nothing”.
I’m with Charlie Brown in not understanding where Lucy was going with all this, though. Maybe she lost the thread herself.
Mark Anderson’sAndertoons for the 28th is Mark Anderson’sAndertoons for the week. Wavehead’s worried about the verbs of both squaring and rounding numbers. Will say it’s a pair of words with contrary alternate meanings that I hadn’t noticed before. I have always taken the use of “square” to reflect, well, if you had a square with sides of size 4, then you’d have a square with area of size 16. The link seems obvious and logical. So on reflection that’s probably not at all where English gets it from. I mean, not to brag or anything but I’ve been speaking English all my life. If I’ve learned anything about it, it’s that the origin is probably something daft like “while Tisquantum [Squanto] was in England he impressed locals with his ability to do arithmetic and his trick of multiplying one number by itself got nicknamed squantuming, which got shortened to squaning to better fit the meter in a music-hall song about him, and a textbook writer in 1704 thought that was a mistake and `corrected’ it to squaring and everyone copied that”. I’m not even going to venture a guess about the etymology of “rounding”.
Marguerite Dabaie and Tom Hart’s Ali’s House for the 28th sets up a homework-help session over algebra. Can’t say where exactly Maisa is going wrong. Her saying “x equals 30 but the train equals” looks like trouble to me. It’s often good practice to start by writing out what are the things in the problem that seem important. And what symbol one wants each to mean. And what one knows about the relationship between these things. It helps clarify why someone would want to do that instead of something else. This is a new comic strip tag and I don’t think I’ve ever had cause to discuss it before.
Hilary Price’s Rhymes With Orange for the 29th is a Rubik’s Cube joke. I’ve counted that as mathematical enough, usually. The different ways that you can rotate parts of the cube form a group. This is something like what I mentioned in the Peanuts discussion. The different rotations you can do can be added to or subtracted from each other, the way numbers can. (Multiplication I’m wary about.)
And now here’s the strip that is unsuitable for reading at work, owing to the appearance of an undressed woman.
I should have got to this yesterday; I don’t know. Something happened. Should be back to normal Sunday.
Bill Rechin’s Crock rerun for the 26th of April does a joke about picking-the-number-in-my-head. There’s more clearly psychological than mathematical content in the strip. It shows off something about what people understand numbers to be, though. It’s easy to imagine someone asked to pick a number choosing “9”. It’s hard to imagine them picking “4,796,034,621,322”, even though that’s just as legitimate a number. It’s possible someone might pick π, or e, but only if that person’s a particular streak of nerd. They’re not going to pick the square root of eleven, or negative eight, or so. There’s thing that are numbers that a person just, offhand, doesn’t think of as numbers.
Mark Anderson’s Andertoons for the 26th sees Wavehead ask about “borrowing” in subtraction. It’s a riff on some of the terminology. Wavehead’s reading too much into the term, naturally. But there are things someone can reasonably be confused about. To say that we are “borrowing” ten does suggest we plan to return it, for example, and we never do that. I’m not sure there is a better term for this turning a digit in one column to adding ten to the column next to it, though. But I admit I’m far out of touch with current thinking in teaching subtraction.
Greg Cravens’s The Buckets for the 26th is kind of a practical probability question. And psychology also, since most of the time we don’t put shirts on wrong. Granted there might be four ways to put a shirt on. You can put it on forwards or backwards, you can put it on right-side-out or inside-out. But there are shirts that are harder to mistake. Collars or a cut around the neck that aren’t symmetric front-to-back make it harder to mistake. Care tags make the inside-out mistake harder to make. We still manage it, but the chance of putting a shirt on wrong is a lot lower than the 75% chance we might naively expect. (New comic tag, by the way.)
Charles Schulz’s Peanuts rerun for the 27th is surely set in mathematics class. The publication date interests me. I’m curious if this is the first time a Peanuts kid has flailed around and guessed “the answer is twelve!” Guessing the answer is twelve would be a Peppermint Patty specialty. But it has to start somewhere.
Knowing nothing about the problem, if I did get the information that my first guess of 12 was wrong, yeah, I’d go looking for 6 or 4 as next guesses, and 12 or 48 after that. When I make an arithmetic mistake, it’s often multiplying or dividing by the wrong number. And 12 has so many factors that they’re good places to look. Subtracting a number instead of adding, or vice-versa, is also common. But there’s nothing in 12 by itself to suggest another place to look, if the addition or subtraction went wrong. It would be in the question which, of course, doesn’t exist.
Maria Scrivan’s Half-Full for the 28th is the Venn Diagram joke for this week. It could include an extra circle for bloggers looking for content they don’t need to feel inspired to write. This one isn’t a new comics tag, which surprises me.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 28th uses the M&oum;bius Strip. It’s an example of a surface that you could just go along forever. There’s nothing topologically special about the M&oum;bius Strip in this regard, though. The mathematician would have as infinitely “long” a résumé if she tied it into a simple cylindrical loop. But the M&oum;bius Strip sounds more exotic, not to mention funnier. Can’t blame anyone going for that instead.
There were nearly a dozen mathematically-themed comic strips among what I’d read, and they almost but not quite split mid-week. Better, they include one of my favorite ever mathematics strips from Charles Schulz’s Peanuts.
Jimmy Halto’s Little Iodine for the 4th of December, 1956 was rerun the 2nd of February. Little Iodine seeks out help with what seems to be story problems. The rate problem — “if it takes one man two hours to plow seven acros, how long will it take five men and a horse to … ” — is a kind I remember being particularly baffling. I think it’s the presence of three numbers at once. It seems easy to go from, say, “if you go two miles in ten minutes, how long will it take to go six miles?” to an answer. To go from “if one person working two hours plows seven acres then how long will five men take to clear fourteen acres” to an answer seems like a different kind of problem altogether. It’s a kind of problem for which it’s even wiser than usual to carefully list everything you need.
Kieran Meehan’s Pros and Cons for the 5th uses a bit of arithmetic. It looks as if it’s meant to be a reminder about following the conclusions of one’s deductive logic. It’s more common to use 1 + 1 equalling 2, or 2 + 2 equalling 4. Maybe 2 times 2 being 4. But then it takes a little turn into numerology, trying to read more meaning into numbers than is wise. (I understand why people should use numerological reasoning, especially given how much mathematicians like to talk up mathematics as descriptions of reality and how older numeral systems used letters to represent words. And that before you consider how many numbers have connotations.)
Mort Walker and Dik Browne’s Hi and Lois for the 10th of August, 1960 was rerun the 6th of February. It’s a counting joke. Babies do have some number sense. At least babies as old as Trixie do, I believe, in that they’re able to detect that something weird is going on when they’re shown, eg, two balls put into a box and four balls coming out. (Also it turns out that stage magicians get called in to help psychologists study just how infants and toddlers understand the world, which is neat.)
It wasn’t just another busy week from Comic Strip Master Command. And a week busy enough for me to split the mathematics comics into two essays. It was one where I recognized one of the panels as one I’d featured before. Multiple times. Some of the comics I feature are in perpetual reruns and don’t have your classic, deep, Peanuts-style decades of archives to draw from. I don’t usually go checking my archives to see if I’ve mentioned a comic before, not unless something about it stands out. So for me to notice I’ve seen this strip repeatedly can mean only one thing: there was something a little bit annoying about it. Recognize it yet? You will.
Hy Eisman’s Popeye for the 7th of January, 2018 is an odd place for mathematics to come in. J Wellington Wimpy regales Popeye with all the intellectual topics he tried to impress his first love with, and “Euclidean postulates in the original Greek” made the cut. And, fair enough. Euclid’s books are that rare thing that’s of important mathematics (or scientific) merit and that a lay person can just pick up and read, even for pleasure. These days we’re more likely to see a division between mathematics writing that’s accessible but unimportant (you know, like, me) or that’s important but takes years of training to understand. Doing it in the original Greek is some arrogant showing-off, though. Can’t blame Carolyn for bailing on someone pulling that stunt.
John Hambrock’s The Brilliant Mind of Edison Lee for the 8th is set in mathematics class. And Edison tries to use a pile of mathematically-tinged words to explain why it’s okay to read a Star Wars book instead of paying attention. Or at least to provide a response the teacher won’t answer. Maybe we can make something out of this by allowing the monetary value of something to be related to its relevance. But if we allow that then Edison’s messed up. I don’t know what quantity is measured by multiplying “every Star Wars book ever written” by “all the movies and merchandise”. But dividing that by the value of the franchise gets … some modest number in peculiar units divided by a large number of dollars. The number value is going to be small. And the dimensions are obviously crazy. Edison needs to pay better attention to the mathematics.
Johnny Hart’s B.C. for the 14th of July, 1960 shows off the famous equation of the 20th century. All part of the comic’s anachronism-comedy chic. The strip reran the 9th of January. “E = mc2” is, correctly, associated with Albert Einstein and some of his important publications of 1905. But the expression does have some curious precursors, people who had worked out the relationship (or something close to it) before Einstein and who didn’t quite know what they had. A short piece from Scientific American a couple years back describes pre-Einstein expressions of the equation from Oliver Heaviside, Henri Poincaré, and Fritz Hasenöhrl. I’m not surprised Poincaré had something close to this; it seems like he spent twenty years almost discovering Relativity. That’s all right; he did enough in dynamical systems that mathematicians aren’t going to forget him.
Jason Chatfield’s Ginger Meggs for the 9th draws my eye just because the blackboard lists “Prime Numbers”. Fair enough place setting, although what’s listed are 1, 3, 5, and 7. These days mathematicians don’t tend to list 1 as a prime number; it’s inconvenient. (A lot of proofs depend on their being exactly one way to factorize a number. But you can always multiply a number by ‘1’ a couple more times without changing its value. So ‘6’ is 3 times 2, but it’s also 3 times 2 times 1, or 3 times 2 times 1 times 1, or 3 times 2 times 1145,388,434,247. You can write around that, but it’s easier to define ‘1’ as not a prime.) But it could be defended. I can’t think any reason to leave ‘2’ off a list of prime numbers, though. I think Chatfield conflated odd and prime numbers. If he’d had a bit more blackboard space we could’ve seen whether the next item was 9 or 11 and that would prove the matter.
Paul Trap’s Thatababy for the 9th uses arithmetic — square roots — as the kind of thing to test whether a computer’s working. Everyone has their little tests like this. My love’s father likes to test whether the computer knows of the band Walk The Moon or of Christine Korsgaard (a prominent philosopher in my love’s specialty). I’ve got a couple words I like to check dictionaries for. Of course the test is only any good if you know what the answer should be, and what’s the actual square root of 3,278? Goodness knows. It’s got to be between 50 (50 squared is 25 hundred) and 60 (60 squared is 36 hundred). Since 3,278 is so much closer 3,600 than 2,500 its square root should be closer to 60 than to 50. So 57-point-something is plausible. Unfortunately square roots don’t lend themselves to the same sorts of tricks from reading the last digit that cube roots do. And 3,278 isn’t a perfect square anyway. Alexa is right on this one. Also about the specific gravity of cobalt, at least if Wikipedia is right and not conspiring with the artificial intelligences on this one. Catch you in 2021.
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.
Comic Strip Master Command hasn’t had many comics exactly on mathematical points the past week. I’ll make do. There are some that are close enough for me, since I like the comics already. And enough of them circle around people being nervous about doing mathematics that I have a title for this edition.
Tony Cochrane’s Agnes for the 24th talks about math anxiety. It’s not a comic strip that will do anything to resolve anyone’s mathematics anxiety. But it’s funny about its business. Agnes usually is; it’s one of the less-appreciated deeply-bizarre comics out there.
Charles Schulz’s Peanuts for the 24th reruns the comic from the 2nd of November, 1970. It has Sally discovering that multiplication is much easier than she imagined. As it is, she’s not in good shape. But if you accept ‘tooty-two’ as another name for ‘four’ and ‘threety-three’ as another name for ‘nine’, why not? And she might do all right in group theory. In that you can select a bunch of things, called ‘elements’, and describe their multiplication to fit anything you like, provided there’s consistency. There could be a four-forty-four if that seems to answer some question.
Hilary Price’s Rhymes with Orange for the 26th is a calculator joke, made explicitly magical. I’m amused but also wonder if those are small wizards or large mushrooms. And it brings up again the question: why do mathematics teachers care about seeing how you got the answer? Who cares, as long as the answer is right? And my answer there is that yeah, sometimes all we care about is the answer. But more often we care about why someone knows the answer is this instead of that. The argument about what makes this answer right — or other answers wrong — should make it possible to tell why. And it often will help inform other problems. Being able to use the work done for one problem to solve others, or better, a whole family of problems, is fantastic. It’s the sort of thing mathematicians naturally try to do.
And now I get caught up again, if briefly, to the mathematically-themed comic strips I can find. I’ve dubbed this one the trapezoid edition because one happens to mention the post that will outlive me.
Todd Clark’s Lola (May 4) is a straightforward joke. Monty’s given his chance of passing mathematics and doesn’t understand the prospect is grim.
Joe Martin’s Willy and Ethel (May 4) shows an astounding feat of mind-reading, or of luck. How amazing it is to draw a number at random from a range depends on many things. It’s less impressive to pick the right number if there are only three possible answers than it is to pick the right number out of ten million possibilities. When we ask someone to pick a number we usually mean a range of the counting numbers. My experience suggests it’s “one to ten” unless some other range is specified. But the other thing affecting how amazing it is is the distribution. There might be ten million possible responses, but if only a few of them are likely then the feat is much less impressive.
The distribution of a random number is the interesting thing about it. The number has some value, yes, and we may not know what it is, but we know how likely it is to be any of the possible values. And good mathematics can be done knowing the distribution of a value of something. The whole field of statistical mechanics is an example of that. James Clerk Maxwell, famous for the equations which describe electromagnetism, used such random variables to explain how the rings of Saturn could exist. It isn’t easy to start solving problems with distributions instead of particular values — I’m not sure I’ve seen a good introduction, and I’d be glad to pass one on if someone can suggest it — but the power it offers is amazing.