The back half of last week’s mathematically themed comic strips aren’t all that deep. They make up for it by being numerous. This is how calculus works, so, good job, Comic Strip Master Command. Here’s what I have for you.
Mark Anderson’s Andertoons for the 20th marks its long-awaited return to these Reading The Comics posts. It’s of the traditional form of the student misunderstanding the teacher’s explanations. Arithmetic edition.
Marty Links’s Emmy Lou for the 20th was a rerun from the 22nd of September, 1976. It’s just a name-drop. It’s not like it matters for the joke which textbook was lost. I just include it because, what the heck, might as well.
Jef Mallett’s Frazz for the 21st uses the form of a story problem. It’s a trick question anyway; there’s really no way the Doppler effect is going to make an ice cream truck’s song unrecognizable, not even at highway speeds. Too distant to hear, that’s a possibility. Also I don’t know how strictly regional this is but the ice cream trucks around here have gone in for interrupting the music every couple seconds with some comical sound effect, like a “boing” or something. I don’t know what this hopes to achieve besides altering the timeline of when the ice cream seller goes mad.
Mark Litzler’s Joe Vanilla for the 21st I already snuck in here last week, in talking about ‘x’. The variable does seem like a good starting point. And, yeah, hypothesis block is kind of a thing. There’s nothing quite like staring at a problem that should be interesting and having no idea where to start. This happens even beyond grade school and the story problems you do then. What to do about it? There’s never one thing. Study it a good while, read about related problems a while. Maybe work on something that seems less obscure a while. It’s very much like writer’s block.
Ryan North’s Dinosaur Comics rerun for the 22nd straddles the borders between mathematics, economics, and psychology. It’s a problem about making forecasts about other people’s behavior. It’s a mystery of game theory. I don’t know a proper analysis for this game. I expect it depends on how many rounds you get to play: if you have a sense of what people typically do, you can make a good guess of what they will do. If everyone gets a single shot to play, all kinds of crazy things might happen.
Jef Mallet’s Frazz gets in again on the 22nd with some mathematics gibberish-talk, including some tossing around of the commutative property. Among other mistakes Caulfield was making here, going from “less is more to therefore more is less” isn’t commutation. Commutation is about binary operations, where you match a pair of things to a single thing. The operation commutes if it never matters what the order of the pair of things is. It doesn’t commute if it ever matters, even a single time, what the order is. Commutativity gets introduced in arithmetic where there are some good examples of the thing. Addition and multiplication commute. Subtraction and division don’t. From there it gets forgotten until maybe eventually it turns up in matrix multiplication, which doesn’t commute. And then it gets forgotten once more until maybe group theory. There, whether operations commute or not is as important a divide as the one between vertebrates and invertebrates. But I understand kids not getting why they should care about commuting. Early on it seems like a longwinded way to say what’s obvious about addition.
Bud Blake’s Tiger rerun for the 23rd starts with a real-world example of your classic story problem. I like the joke in it, and I also like Hugo’s look of betrayal and anger in the second panel. A spot of expressive art will do so good for a joke.
We come now almost to the end of the Summer 2017 A To Z. Possibly also the end of all these A To Z sequences. Gaurish of, For the love of Mathematics, proposed that I talk about the obvious logical choice. The last promising thing I hadn’t talked about. I have no idea what to do for future A To Z’s, if they’re even possible anymore. But that’s a problem for some later time.
Some good advice that I don’t always take. When starting a new problem, make a list of all the things that seem likely to be relevant. Problems that are worth doing are usually about things. They’ll be quantities like the radius or volume of some interesting surface. The amount of a quantity under consideration. The speed at which something is moving. The rate at which that speed is changing. The length something has to travel. The number of nodes something must go across. Whatever. This all sounds like stuff from story problems. But most interesting mathematics is from a story problem; we want to know what this property is like. Even if we stick to a purely mathematical problem, there’s usually a couple of things that we’re interested in and that we describe. If we’re attacking the four-color map theorem, we have the number of territories to color. We have, for each territory, the number of territories that touch it.
Next, select a name for each of these quantities. Write it down, in the table, next to the term. The volume of the tank is ‘V’. The radius of the tank is ‘r’. The height of the tank is ‘h’. The fluid is flowing in at a rate ‘r’. The fluid is flowing out at a rate, oh, let’s say ‘s’. And so on. You might take a moment to go through and think out which of these variables are connected to which other ones, and how. Volume, for example, is surely something to do with the radius times something to do with the height. It’s nice to have that stuff written down. You may not know the thing you set out to solve, but you at least know you’ve got this under control.
I recommend this. It’s a good way to organize your thoughts. It establishes what things you expect you could know, or could want to know, about the problem. It gives you some hint how these things relate to each other. It sets you up to think about what kinds of relationships you figure to study when you solve the problem. It gives you a lifeline, when you’re lost in a sea of calculation. It’s reassurance that these symbols do mean something. Better, it shows what those things are.
I don’t always do it. I have my excuses. If I’m doing a problem that’s very like one I’ve already recently done, the things affecting it are probably the same. The names to give these variables are probably going to be about the same. Maybe I’ll make a quick sketch to show how the parts of the problem relate. If it seems like less work to recreate my thoughts than to write them down, I skip writing them down. Not always good practice. I tell myself I can always go back and do things the fully right way if I do get lost. So far that’s been true.
So, the names. Suppose I am interested in, say, the length of the longest rod that will fit around this hallway corridor. Then I am in a freshman calculus book, yes. Fine. Suppose I am interested in whether this pinball machine can be angled up the flight of stairs that has a turn in it Then I will measure things like the width of the pinball machine. And the width of the stairs, and of the landing. I will measure this carefully. Pinball machines are heavy and there are many hilarious sad stories of people wedging them into hallways and stairwells four and a half stories up from the street. But: once I have identified, say, ‘width of pinball machine’ as a quantity of interest, why would I ever refer to it as anything but?
This is no dumb question. It is always dangerous to lose the link between the thing we calculate and the thing we are interested in. Without that link we are less able to notice mistakes in either our calculations or the thing we mean to calculate. Without that link we can’t do a sanity check, that reassurance that it’s not plausible we just might fit something 96 feet long around the corner. Or that we estimated that we could fit something of six square feet around the corner. It is common advice in programming computers to always give variables meaningful names. Don’t write ‘T’ when ‘Total’ or, better, ‘Total_Value_Of_Purchase’ is available. Why do we disregard this in mathematics, and switch to ‘T’ instead?
First reason is, well, try writing this stuff out. Your hand (h) will fall off (foff) in about fifteen minutes, twenty seconds. (15′ 20”). If you’re writing a program, the programming environment you have will auto-complete the variable after one or two letters in. Or you can copy and paste the whole name. It’s still good practice to leave a comment about what the variable should represent, if the name leaves any reasonable ambiguity.
Another reason is that sure, we do specific problems for specific cases. But a mathematician is naturally drawn to thinking of general problems, in abstract cases. We see something in common between the problem “a length and a quarter of the length is fifteen feet; what is the length?” and the problem “a volume plus a quarter of the volume is fifteen gallons; what is the volume?”. That one is about lengths and the other about volumes doesn’t concern us. We see a saving in effort by separating the quantity of a thing from the kind of the thing. This restores danger. We must think, after we are done calculating, about whether the answer could make sense. But we can minimize that, we hope. At the least we can check once we’re done to see if our answer makes sense. Maybe even whether it’s right.
For centuries, as the things we now recognize as algebra developed, we would use words. We would talk about the “thing” or the “quantity” or “it”. Some impersonal name, or convenient pronoun. This would often get shortened because anything you write often you write shorter. “Re”, perhaps. In the late 16th century we start to see the “New Algebra”. Here mathematics starts looking like … you know … mathematics. We start to see stuff like “addition” represented with the + symbol instead of an abbreviation for “addition” or a p with a squiggle over it or some other shorthand. We get equals signs. You start to see decimals and exponents. And we start to see letters used in place of numbers whose value we don’t know.
There are a couple kinds of “numbers whose value we don’t know”. One is the number whose value we don’t know, but hope to learn. This is the classic variable we want to solve for. Another kind is the number whose value we don’t know because we don’t care. I mean, it has some value, and presumably it doesn’t change over the course of our problem. But it’s not like our work will be so different if, say, the tank is two feet high rather than four.
Is there a problem? If we pick our letters to fit a specific problem, no. Presumably all the things we want to describe have some clear name, and some letter that best represents the name. It’s annoying when we have to consider, say, the pinball machine width and the corridor width. But we can work something out.
But what about general problems?
Is an easy problem to solve?
If we want to figure what ‘m’ is, yes. Similarly ‘y’. If we want to know what ‘b’ is, it’s tedious, but we can do that. If we want to know what ‘e’ is? Run and hide, that stuff is crazy. If you have to, do it numerically and accept an estimate. Don’t try figuring what that is.
And so we’ve developed conventions. There are some letters that, except in weird circumstances, are coefficients. They’re numbers whose value we don’t know, but either don’t care about or could look up. And there are some that, by default, are variables. They’re the ones whose value we want to know.
These conventions started forming, as mentioned, in the late 16th century. François Viète here made a name that lasts to mathematics historians at least. His texts described how to do algebra problems in the sort of procedural methods that we would recognize as algebra today. And he had a great idea for these letters. Use the whole alphabet, if needed. Use the consonants to represent the coefficients, the numbers we know but don’t care what they are. Use the vowels to represent the variables, whose values we want to learn. So he would look at that equation and see right away: it’s a terrible mess. (I exaggerate. He doesn’t seem to have known the = sign, and I don’t know offhand when ‘log’ and ‘cos’ became common. But suppose the rest of the equation were translated into his terminology.)
It’s not a bad approach. Besides the mnemonic value of consonant-coefficient, vowel-variable, it’s true that we usually have fewer variables than anything else. The more variables in a problem the harder it is. If someone expects you to solve an equation with ten variables in it, you’re excused for refusing. So five or maybe six or possibly seven choices for variables is plenty.
But it’s not what we settled on. René Descartes had a better idea. He had a lot of them, but here’s one. Use the letters at the end of the alphabet for the unknowns. Use the letters at the start of the alphabet for coefficients. And that is, roughly, what we’ve settled on. In my example nightmare equation, we’d suppose ‘y’ to probably be the variable we want to solve for.
And so, and finally, x. It is almost the variable. It says “mathematics” in only two strokes. Even π takes more writing. Descartes used it. We follow him. It’s way off at the end of the alphabet. It starts few words, very few things, almost nothing we would want to measure. (Xylem … mass? Flow? What thing is the xylem anyway?) Even mathematical dictionaries don’t have much to say about it. The letter transports almost no connotations, no messy specific problems to it. If it suggests anything, it suggests the horizontal coordinate in a Cartesian system. It almost is mathematics. It signifies nothing in itself, but long use has given it an identity as the thing we hope to learn by study.
And pirate treasure maps. I don’t know when ‘X’ became the symbol of where to look for buried treasure. My casual reading suggests “never”. Treasure maps don’t really exist. Maps in general don’t work that way. Or at least didn’t before cartoons. X marking the spot seems to be the work of Robert Louis Stevenson, renowned for creating a fanciful map and then putting together a book to justify publishing it. (I jest. But according to Simon Garfield’s On The Map: A Mind-Expanding Exploration of the Way The World Looks, his map did get lost on the way to the publisher, and he had to re-create it from studying the text of Treasure Island. This delights me to no end.) It makes me wonder if Stevenson was thinking of x’s service in mathematics. But the advantages of x as a symbol are hard to ignore. It highlights a point clearly. It’s fast to write. Its use might be coincidence.
But it is a letter that does a needed job really well.
If there was a theme this week, it was puzzles. So many strips had little puzzles to work out. You’ll see. Thank you.
Bill Amend’s FoxTrot for the 30th of April tries to address my loss of Jumble panels. Thank you, whoever at Comic Strip Master Command passed along word of my troubles. I won’t spoil your fun. As sometimes happens with a Jumble you can work out the joke punchline without doing any of the earlier ones. 64 in binary would be written 1000000. And from this you know what fits in all the circles of the unscrambled numbers. This reduces a lot of the scrambling you have to do: just test whether 341 or 431 is a prime number. Check whether 8802, 8208, or 2808 is divisible by 117. The integer cubed you just have to keep trying possibilities. But only one combination is the cube of an integer. The factorial of 12, just, ugh. At least the circles let you know you’ve done your calculations right.
Steve McGarry’s activity feature Kidtown for the 30th plays with numbers some. And a puzzle that’ll let you check how well you can recognize multiles of four that are somewhere near one another. You can use diagonals too; that’s important to remember.
Mac King and Bill King’s Magic in a Minute feature for the 30th is also a celebration of numerals. Enjoy the brain teaser about why the encoding makes sense. I don’t believe the hype about NASA engineers needing days to solve a puzzle kids got in minutes. But if it’s believable, is it really hype?
Marty Links’s Emmy Lou from the 29th of October, 1963 was rerun the 2nd of May. It’s a reminder that mathematics teachers of the early 60s also needed something to tape to their doors.
Mark Litzler’s Joe Vanilla for the 2nd name-drops the Null Hypothesis. I’m not sure what Litzler is going for exactly. The Null Hypothesis, though, comes to us from statistics and from inference testing. It turns up everywhere when we sample stuff. It turns up in medicine, in manufacturing, in psychology, in economics. Everywhere we might see something too complicated to run the sorts of unambiguous and highly repeatable tests that physics and chemistry can do — things that are about immediately practical questions — we get to testing inferences. What we want to know is, is this data set something that could plausibly happen by chance? Or is it too far out of the ordinary to be mere luck? The Null Hypothesis is the explanation that nothing’s going on. If your sample is weird in some way, well, everything is weird. What’s special about your sample? You hope to find data that will let you reject the Null Hypothesis, showing that the data you have is so extreme it just can’t plausibly be chance. Or to conclude that you fail to reject the Null Hypothesis, showing that the data is not so extreme that it couldn’t be chance. We don’t accept the Null Hypothesis. We just allow that more data might come in sometime later.
I don’t know what Litzler is going for with this. I feel like I’m missing a reference and I’ll defer to a finance blogger’s Reading the Comics post.
Greg Evans’s Luann Againn for the 28th of February — reprinting the strip from the same day in 1989 — uses a bit of arithmetic as generic homework. It’s an interesting change of pace that the mathematics homework is what keeps one from sleep. I don’t blame Luann or Puddles for not being very interested in this, though. Those sorts of complicated-fraction-manipulation problems, at least when I was in middle school, were always slogs of shuffling stuff around. They rarely got to anything we’d like to know.
Jef Mallett’s Frazz for the 1st of March is one of those little revelations that statistics can give one. Myself, I was always haunted by the line in Carl Sagan’s Cosmos about how, in the future, with the Sun ageing and (presumably) swelling in size and heat, the Earth would see one last perfect day. That there would most likely be quite fine days after that didn’t matter, and that different people might disagree on what made a day perfect didn’t matter. Setting out the idea of a “perfect day” and realizing there would someday be a last gave me chills. It still does.
Richard Thompson’s Poor Richard’s Almanac for the 1st and the 2nd of March have appeared here before. But I like the strip so I’ll reuse them too. They’re from the strip’s guide to types of Christmas trees. The Cubist Fur is described as “so asymmetrical it no longer inhabits Euclidean space”. Properly neither do we, but we can’t tell by eye the difference between our space and a Euclidean space. “Non-Euclidean” has picked up connotations of being so bizarre or even horrifying that we can’t hope to understand it. In practice, it means we have to go a little slower and think about, like, what would it look like if we drew a triangle on a ball instead of a sheet of paper. The Platonic Fir, in the 2nd of March strip, looks like a geometry diagram and I doubt that’s coincidental. It’s very hard to avoid thoughts of Platonic Ideals when one does any mathematics with a diagram. We know our drawings aren’t very good triangles or squares or circles especially. And three-dimensional shapes are worse, as see every ellipsoid ever done on a chalkboard. But we know what we mean by them. And then we can get into a good argument about what we mean by saying “this mathematical construct exists”.
Mark Litzler’s Joe Vanilla for the 3rd uses a chalkboard full of mathematics to represent the deep thinking behind a silly little thing. I can’t make any of the symbols out to mean anything specific, but I do like the way it looks. It’s quite well-done in looking like the shorthand that, especially, physicists would use while roughing out a problem. That there are subscripts with forms like “12” and “22” with a bar over them reinforces that. I would, knowing nothing else, expect this to represent some interaction between particles 1 and 2, and 2 with itself, and that the bar means some kind of complement. This doesn’t mean much to me, but with luck, it means enough to the scientist working it out that it could be turned into a coherent paper.
Bill Holbrook’s On The Fastrack is this week about the wedding of the accounting-minded Fi. And she’s having last-minute doubts, which is why the strip of the 3rd brings in irrational and anthropomorphized numerals. π gets called in to serve as emblematic of the irrational numbers. Can’t fault that. I think the only more famously irrational number is the square root of two, and π anthropomorphizes more easily. Well, you can draw an established character’s face onto π. The square root of 2 is, necessarily, at least two disconnected symbols and you don’t want to raise distracting questions about whether the root sign or the 2 gets the face.
That said, it’s a lot easier to prove that the square root of 2 is irrational. Even the Pythagoreans knew it, and a bright child can follow the proof. A really bright child could create a proof of it. To prove that π is irrational is not at all easy; it took mathematicians until the 19th century. And the best proof I know of the fact does it by a roundabout method. We prove that if a number (other than zero) is rational then the tangent of that number must be irrational, and vice-versa. And the tangent of π/4 is 1, so therefore π/4 must be irrational, so therefore π must be irrational. I know you’ll all trust me on that argument, but I wouldn’t want to sell it to a bright child.
Holbrook continues the thread on the 4th, extends the anthropomorphic-mathematics-stuff to call people variables. There’s ways that this is fair. We use a variable for a number whose value we don’t know or don’t care about. A “random variable” is one that could take on any of a set of values. We don’t know which one it does, in any particular case. But we do know — or we can find out — how likely each of the possible values is. We can use this to understand the behavior of systems even if we never actually know what any one of it does. You see how I’m going to defend this metaphor, then, especially if we allow that what people are likely or unlikely to do will depend on context and evolve in time.