## No, You Can’t Say What 6/2(1+2) Equals

I am made aware that a section of Twitter argues about how to evaluate an expression. There may be more than one of these going around, but the expression I’ve seen is:

$6 \div 2\left(1 + 2\right) =$

Many people feel that the challenge is knowing the order of operations. This is reasonable. That is, that to evaluate arithmetic, you evaluate terms inside parentheses first. Then terms within exponentials. Then multiplication and division. Then addition and subtraction. This is often abbreviated as PEMDAS, and made into a mnemonic like “Please Excuse My Dear Aunt Sally”.

That is fine as far as it goes. Many people likely start by adding the 1 and 2 within the parentheses, and that’s fair. Then they get:

$6 \div 2(3) =$

Putting two quantities next to one another, as the 2 and the (3) are, means to multiply them. And then comes the disagreement: does this mean take $6\div 2$ and multiply that by 3, in which case the answer is 9? Or does it mean take 6 divided by $2\cdot 3$, in which case the answer is 1?

And there is the trick. Depending on which way you choose to parse these instructions you get different answers. But you don’t get to do that, not and have arithmetic. So the answer is that this expression has no answer. The phrasing is ambiguous and can’t be resolved.

I’m aware there are people who reject this answer. They picked up along the line somewhere a rule like “do multiplication and division from left to right”. And a similar rule for addition and subtraction. This is wrong, but understandable. The left-to-right “rule” is a decent heuristic, a guide to how to attack a problem too big to do at once. The rule works because multiplication-and-division associates. The quantity a-times-b, multiplied by c, has to be the same number as the quantity a multiplied by the quantity b-times-c. The rule also works for addition-and-subtraction because addition associates too. The quantity a-plus-b, plus the quantity c, has to be the same as the quantity a plus the quantity b-plus-c.

This left-to-right “rule”, though, just helps you evaluate a meaningful expression. It would be just as valid to do all the multiplications-and-divisions from right-to-left. If you get different values working left-to-right from right-to-left, you have a meaningless expression.

But you also start to see why mathematicians tend to avoid the $\div$ symbol. We understand, for example, $a \div b$ to mean $a \cdot \frac{1}{b}$. Carry that out and then there’s no ambiguity about

$6 \cdot \frac{1}{2} \cdot 3 =$

I understand the desire to fix an ambiguity. Believe me. I’m a know-it-all; I only like ambiguities that enable logic-based jokes. (“Would you like ice cream or cake?” “Yes.”) But the rules that could remove the ambiguity in $6\div 2(1 + 2)$ also remove associativity from multiplication. Once you do that, you’re not doing arithmetic anymore. Resist the urge.

(And the mnemonic is a bit dangerous. We can say division has the same priority as multiplication, but we also say “multiplication” first. I bet you can construct an ambiguous expression which would mislead someone who learned Please Excuse Dear Miss Sally Andrews.)

And now a qualifier: computer languages will often impose doing a calculation in some order. Usually left-to-right. The microchips doing the work need to have some instructions. Spotting all possible ambiguous phrasings ahead of time is a challenge. But we accept our computers doing not-quite-actual-arithmetic. They’re able to do not-quite-actual-arithmetic much faster and more reliably than we can. This makes the compromise worthwhile. We need to remember the difference between what the computer does and the calculation we intend.

And another qualifier: it is possible to do interesting mathematics with operations that aren’t associative. But if you are it’s in your research as a person with a postgraduate degree in mathematics. It’s possible it might fit in social media, but I would be surprised. It won’t draw great public attention, anyway.

## Reading the Comics, November 27, 2018: Multiplication Edition

Last week Comic Strip Master Command sent out just enough on-theme comics for two essays, the way I do them these days. The first half has some multiplication in two of the strips. So that’s enough to count as a theme for me.

Aaron Neathery’s Endtown for the 26th depicts a dreary, boring school day by using arithmetic. A lot of times tables. There is some credible in-universe reason to be drilling on multiplication like this. The setting is one where the characters can’t expect to have computers available. That granted, I’m not sure there’s a point to going up to memorizing four times 27. Going up to twelve-times seems like enough for common uses. For multiplying two- and longer-digit numbers together we usually break the problem up into a string of single-digit multiplications.

There are a handful of bigger multiplications that can make your life easier to know, like how four times 25 is 100. Or three times 33 is pretty near 100. But otherwise? … Of course, the story needs the class to do something dull and seemingly pointless. Going deep into multiplication tables communicates that to the reader quickly.

Thaves’s Frank and Ernest for the 26th is a spot of wordplay. Also a shout-out to my friends who record mathematics videos for YouTube. It is built on the conflation between the ideas of something multiplying and the amount of something growing. It’s easy to see where the idea comes from; just keep hitting ‘x 2’ on a calculator and the numbers grow excitingly fast. You get even more exciting results with ‘x 3’ or ‘x π’. But multiplying by 1 is still multiplication. As is multiplying by a number smaller than 1. Including negative numbers. That doesn’t hurt the joke any. That multiplying two things together doesn’t necessarily give you something larger is a consideration when you’re thinking rigorously about what multiplication can do. It doesn’t have to be part of normal speech.

John Hambrock’s The Brilliant Mind of Edison Lee for the 27th uses the form of a word problem to show off Edison’s gluttony. Edison tries to present it as teaching. We all have rationalizations for giving in to our appetites.

Nate Frakes’s Break of Day for the 27th is the anthropomorphic numerals joke for the week. I don’t know that there’s anything in the other numerals being odds rather than evens, or a mixture of odds and evens. It might just be that they needed to be anything but 1.

All of my regular Reading the Comics posts should all be at this link. The next in my Fall 2018 Mathematics A To Z glossary should be posted Tuesday. I’m glad for it if you do come around and read again.

## What You Need To Pass This Class. Also: It’s Algebra, Uncle Fletcher

The end of the (US) semester snuck up on me but, in my defense, I’m not teaching this semester. If you know someone who needs me to teach, please leave me a note. But as a service for people who are just trying to figure out exactly how much studying they need to do for their finals, knock it off. You’re not playing a video game. It’s not like you can figure out how much effort it takes to get an 83.5 on the final and then put the rest of your energy into your major’s classes.

But it’s a question people ask, and keep asking, so here’s my answers. This essay describes exactly how to figure out what you need, given whatever grade you have and whatever extra credit you have and whatever the weighting of the final exam is and all that. That might be more mechanism than you need. If you’re content with an approximate answer, here’s some tables for common finals weightings, and a selection of pre-final grades.

For those not interested in grade-grubbing, here’s some old-time radio. Vic and Sade was a longrunning 15-minute morning radio program written with exquisite care by Paul Rhymer. It’s not going to be to everyone’s taste. But if it is yours, it’s going to be really yours: a tiny cast of people talking not quite past one another while respecting the classic Greek unities. Part of the Overnightscape Underground is the Vic and Sadecast, which curates episodes of the show, particularly trying to explain the context of things gone by since 1940. This episode, from October 1941, is aptly titled “It’s Algebra, Uncle Fletcher”. Neither Vic nor Sade are in the episode, but their son Rush and Uncle Fletcher are. And they try to work through high school algebra problems. I’m tickled to hear Uncle Fletcher explaining mathematics homework. I hope you are too.

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

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

Hector D Cantu and Carlos Castellanos’s Baldo for the 17th has Gracie quizzed about percentages of small prices, apparently as a test of her arithmetic. Her aunt has other ideas in mind. It’s hard to dispute that this is mathematics people use in real life. The commenters on GoComics got into an argument about whether Gracie gave the right answers, though. That is, not that 20 percent of $5.95 is anything about$1.19. But did Tia Carmen want to know what 20 percent of $5.95, or did she want to know what$5.95 minus 20 percent of that price was? Should Gracie have answered $4.76 instead? It took me a bit to understand what the ambiguity was, but now that I see it, I’m glad I didn’t write a multiple-choice test with both$1.19 and $4.76 as answers. I’m not sure how to word the questions to avoid ambiguity yet still sound like something one of the hew-mons might say. Dan Thompson’s Brevity for the 19th uses the blackboard and symbols on it as how a mathematician would prove something. In this case, love. Arithmetic’s a good visual way of communicating the mathematician at work here. I don’t think a mathematician would try arguing this in arithmetic, though. I mean if we take the premise at face value. I’d expect an argument in statistics, so, a mathematician showing various measures of … feelings or something. And tests to see whether it’s plausible this cluster of readings could come out by some reason other than love. If that weren’t used, I’d expect an argument in propositional logic. And that would have long strings of symbols at work, but they wouldn’t look like arithmetic. They look more like Ancient High Martian. Just saying. Reza Farazmand’s Poorly Drawn Lines for the 20th you maybe already saw going around your social media. It’s well-designed for that. Also for grad students’ office doors. Dave Coverly’s Speed Bump for the 20th is designed with crossover appeal in mind and I wonder if whoever does Reading the Comics for English Teacher Jokes is running this same strip in their collection for the week. Darrin Bell’s Candorville for the 21st sees Lemont worry that he’s forgotten how to do long division. And, fair enough: any skill you don’t use in long enough becomes stale, whether it’s division or not. You have to keep in practice and, in time, have to decide what you want to keep in practice about. (That said, I have a minor phobia about forgetting how to prove the Contraction Mapping Theorem, as several professors in grad school stressed how it must always be possible to give a coherent proof of that, even if you’re startled awake in the middle of the night by your professor.) Me, I would begin by estimating what 4,858.8 divided by 297.492 should be. 297.492 is very near 300. And 4,858.8 is a little over 4800. And that’s suggestive because it’s obvious that 48 divided by 3 is 16. Well, it’s obvious to me. So I would expect the answer to be “a little more than 16” and, indeed, it’s about 16.3. (Don’t read the comments on GoComics. There’s some slide-rule-snobbishness, and some snark about the uselessness of the skill or the dumbness of Facebook readers, and one comment about too many people knowing how to multiply by someone who’s reading bad population-bomb science fiction of the 70s.) ## Reading the Comics, October 7, 2017: Rerun Comics Edition The most interesting mathematically-themed comic strips from last week were also reruns. So be it; at least I have an excuse to show a 1931-vintage comic. Also, after discovering my old theme didn’t show the category of essay I was posting, I did literally minutes of search for a new theme that did. And that showed tags. And that didn’t put a weird color behind LaTeX inline equations. So I’m using the same theme as my humor blog does, albeit with a different typeface, and we’ll hope that means I don’t post stuff to the wrong blog. As it is I start posting something to the wrong place about once every twenty times. All I want is a WordPress theme with all the good traits of the themes I look at and none of the drawbacks; why is that so hard to get? Elzie Segar’s Thimble Theatre rerun for the 5th originally ran the 25th of April, 1931. It’s just a joke about Popeye not being good at bookkeeping. In the story, Popeye’s taking the$50,000 reward from his last adventure and opened a One-Way Bank, giving people whatever money they say they need. And now you understand how the first panel of the last row has several jokes in it. The strip is partly a joke about Popeye being better with stuff he can hit than anything else, of course. I wonder if there’s an old stereotype of sailors being bad at arithmetic. I remember reading about pirate crews that, for example, not-as-canny-as-they-think sailors would demand a fortieth or a fiftieth of the prizes as their pay, instead of a mere thirtieth. But it’s so hard to tell what really happened and what’s just a story about the stupidity of people. Marginal? Maybe, but I’m a Popeye fan and this is my blog, so there.

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

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

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

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

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

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

## Reading the Comics, August 15, 2017: Cake Edition

It was again a week just busy enough that I’m comfortable splitting the Reading The Comments thread into two pieces. It’s also a week that made me think about cake. So, I’m happy with the way last week shaped up, as far as comic strips go. Other stuff could have used a lot of work Let’s read.

Stephen Bentley’s Herb and Jamaal rerun for the 13th depicts “teaching the kids math” by having them divide up a cake fairly. I accept this as a viable way to make kids interested in the problem. Cake-slicing problems are a corner of game theory as it addresses questions we always find interesting. How can a resource be fairly divided? How can it be divided if there is not a trusted authority? How can it be divided if the parties do not trust one another? Why do we not have more cake? The kids seem to be trying to divide the cake by volume, which could be fair. If the cake slice is a small enough wedge they can likely get near enough a perfect split by ordinary measures. If it’s a bigger wedge they’d need calculus to get the answer perfect. It’ll be well-approximated by solids of revolution. But they likely don’t need perfection.

This is assuming the value of the icing side is not held in greater esteem than the bare-cake sides. This is not how I would value the parts of the cake. They’ll need to work something out about that, too.

Mac King and Bill King’s Magic in a Minute for the 13th features a bit of numerical wizardry. That the dates in a three-by-three block in a calendar will add up to nine times the centered date. Why this works is good for a bit of practice in simplifying algebraic expressions. The stunt will be more impressive if you can multiply by nine in your head. I’d do that by taking ten times the given date and then subtracting the original date. I won’t say I’m fond of the idea of subtracting 23 from 230, or 17 from 170. But a skilled performer could do something interesting while trying to do this subtraction. (And if you practice the trick you can get the hang of the … fifteen? … different possible answers.)

Bill Amend’s FoxTrot rerun for the 14th mentions mathematics. Young nerd Jason’s trying to get back into hand-raising form. Arithmetic has considerable advantages as a thing to practice answering teachers. The questions have clear, definitely right answers, that can be worked out or memorized ahead of time, and can be asked in under half a panel’s word balloon space. I deduce the strip first ran the 21st of August, 2006, although that image seems to be broken.

Ed Allison’s Unstrange Phenomena for the 14th suggests changes in the definition of the mile and the gallon to effortlessly improve the fuel economy of cars. As befits Allison’s Dadaist inclinations the numbers don’t work out. As it is, if you defined a New Mile of 7,290 feet (and didn’t change what a foot was) and a New Gallon of 192 fluid ounces (and didn’t change what an old fluid ounce was) then a 20 old-miles-per-old-gallon car would come out to about 21.7 new-miles-per-new-gallon. Commenter Del_Grande points out that if the New Mile were 3,960 feet then the calculation would work out. This inspires in me curiosity. Did Allison figure out the numbers that would work and then make a mistake in the final art? Or did he pick funny-looking numbers and not worry about whether they made sense? No way to tell from here, I suppose. (Allison doesn’t mention ways to get in touch on the comic’s About page and I’ve only got the weakest links into the professional cartoon community.)

Patrick Roberts’s Todd the Dinosaur for the 15th mentions long division as the stuff of nightmares. So it is. I guess MathWorld and Wikipedia endorse calling 128 divided by 4 long division, although I’m not sure I’m comfortable with that. This may be idiosyncratic; I’d thought of long division as where the divisor is two or more digits. A three-digit number divided by a one-digit one doesn’t seem long to me. I’d just think that was division. I’m curious what readers’ experiences have been.

## Reading the Comics, August 9, 2017: Pets Doing Mathematics Edition

I had just enough comic strips to split this week’s mathematics comics review into two pieces. I like that. It feels so much to me like I have better readership when I have many days in a row with posting something, however slight. The A to Z is good for three days a week, and if comic strips can fill two of those other days then I get to enjoy a lot of regular publication days. … Though last week I accidentally set the Sunday comics post to appear on Monday, just before the A To Z post. I’m curious how that affected my readers. That nobody said anything is ominous.

Niklas Eriksson’s Carpe Diem for the 7th of August uses mathematics as the signifier for intelligence. I’m intrigued by how the joke goes a little different: while the border collies can work out the mechanics of a tossed stick, they haven’t figured out what the point of fetch is. But working out people’s motivations gets into realms of psychology and sociology and economics. There the mathematics might not be harder, but knowing that one is calculating a relevant thing is. (Eriksson’s making a running theme of the intelligence of border collies.)

Nicole Hollander’s Sylvia rerun for the 7th tosses off a mention that “we’re the first generation of girls who do math”. And that therefore there will be a cornucopia of new opportunities and good things to come to them. There’s a bunch of social commentary in there. One is the assumption that mathematics skill is a liberating thing. Perhaps it is the gloom of the times but I doubt that an oppressed group developing skills causes them to be esteemed. It seems more likely to me to make the skills become devalued. Social justice isn’t a matter of good exam grades.

Then, too, it’s not as though women haven’t done mathematics since forever. Every mathematics department on a college campus has some faded posters about Emmy Noether and Sofia Kovalevskaya and maybe Sophie Germaine. Probably high school mathematics rooms too. Again perhaps it’s the gloom of the times. But I keep coming back to the goddess’s cynical dismissal of all this young hope.

Mort Walker and Dik Browne’s Hi and Lois for the 10th of February, 1960 and rerun the 8th portrays arithmetic as a grand-strategic imperative. Well, it means education as a strategic imperative. But arithmetic is the thing Dot uses. I imagine because it is so easy to teach as a series of trivia and quiz about. And it fits in a single panel with room to spare.

Paul Trap’s Thatababy for the 8th is not quite the anthropomorphic-numerals joke of the week. It circles around that territory, though, giving a couple of odd numbers some personality.

Brian Anderson’s Dog Eat Doug for the 9th finally justifies my title for this essay, as cats ponder mathematics. Well, they ponder quantum mechanics. But it’s nearly impossible to have a serious thought about that without pondering its mathematics. This doesn’t mean calculation, mind you. It does mean understanding what kinds of functions have physical importance. And what kinds of things one can do to functions. Understand them and you can discuss quantum mechanics without being mathematically stupid. And there’s enough ways to be stupid about quantum mechanics that any you can cut down is progress.

## The Summer 2017 Mathematics A To Z: Arithmetic

And now as summer (United States edition) reaches its closing months I plunge into the fourth of my A To Z mathematics-glossary sequences. I hope I know what I’m doing! Today’s request is one of several from Gaurish, who’s got to be my top requester for mathematical terms and whom I thank for it. It’s a lot easier writing these things when I don’t have to think up topics. Gaurish hosts a fine blog, For the love of Mathematics, which you might consider reading.

# Arithmetic.

Arithmetic is what people who aren’t mathematicians figure mathematicians do all day. I remember in my childhood a Berenstain Bears book about people’s jobs. Its mathematician was an adorable little bear adding up sums on the chalkboard, in an observatory, on the Moon. I liked every part of this. I wouldn’t say it’s the whole reason I became a mathematician but it did made the prospect look good early on.

People who aren’t mathematicians are right. At least, the bulk of what mathematics people do is arithmetic. If we work by volume. Arithmetic is about the calculations we do to evaluate or solve polynomials. And polynomials are everything that humans find interesting. Arithmetic is adding and subtracting, of multiplication and division, of taking powers and taking roots. Arithmetic is changing the units of a thing, and of breaking something into several smaller units, or of merging several smaller units into one big one. Arithmetic’s role in commerce and in finance must overwhelm the higher mathematics. Higher mathematics offers cohomologies and Ricci tensors. Arithmetic offers a budget.

This is old mathematics. There’s evidence of humans twenty thousands of years ago recording their arithmetic computations. My understanding is the evidence is ambiguous and interpretations vary. This seems fair. I assume that humans did such arithmetic then, granting that I do not know how to interpret archeological evidence. The thing is that arithmetic is older than humans. Animals are able to count, to do addition and subtraction, perhaps to do harder computations. (I crib this from The Number Sense:
How the Mind Creates Mathematics
, by Stanislas Daehaene.) We learn it first, refining our rough instinctively developed sense to something rigorous. At least we learn it at the same time we learn geometry, the other branch of mathematics that must predate human existence.

The primality of arithmetic governs how it becomes an adjective. We will have, for example, the “arithmetic progression” of terms in a sequence. This is a sequence of numbers such as 1, 3, 5, 7, 9, and so on. Or 4, 9, 14, 19, 24, 29, and so on. The difference between one term and its successor is the same as the difference between the predecessor and this term. Or we speak of the “arithmetic mean”. This is the one found by adding together all the numbers of a sample and dividing by the number of terms in the sample. These are important concepts, useful concepts. They are among the first concepts we have when we think of a thing. Their familiarity makes them easy tools to overlook.

Consider the Fundamental Theorem of Arithmetic. There are many Fundamental Theorems; that of Algebra guarantees us the number of roots of a polynomial equation. That of Calculus guarantees us that derivatives and integrals are joined concepts. The Fundamental Theorem of Arithmetic tells us that every whole number greater than one is equal to one and only one product of prime numbers. If a number is equal to (say) two times two times thirteen times nineteen, it cannot also be equal to (say) five times eleven times seventeen. This may seem uncontroversial. The budding mathematician will convince herself it’s so by trying to work out all the ways to write 60 as the product of prime numbers. It’s hard to imagine mathematics for which it isn’t true.

But it needn’t be true. As we study why arithmetic works we discover many strange things. This mathematics that we know even without learning is sophisticated. To build a logical justification for it requires a theory of sets and hundreds of pages of tight reasoning. Or a theory of categories and I don’t even know how much reasoning. The thing that is obvious from putting a couple objects on a table and then a couple more is hard to prove.

As we continue studying arithmetic we start to ponder things like Goldbach’s Conjecture, about even numbers (other than two) being the sum of exactly two prime numbers. This brings us into number theory, a land of fascinating problems. Many of them are so accessible you could pose them to a person while waiting in a fast-food line. This befits a field that grows out of such simple stuff. Many of those are so hard to answer that no person knows whether they are true, or are false, or are even answerable.

And it splits off other ideas. Arithmetic starts, at least, with the counting numbers. It moves into the whole numbers and soon all the integers. With division we soon get rational numbers. With roots we soon get certain irrational numbers. A close study of this implies there must be irrational numbers that must exist, at least as much as “four” exists. Yet they can’t be reached by studying polynomials. Not polynomials that don’t already use these exotic irrational numbers. These are transcendental numbers. If we were to say the transcendental numbers were the only real numbers we would be making only a very slight mistake. We learn they exist by thinking long enough and deep enough about arithmetic to realize there must be more there than we realized.

Thought compounds thought. The integers and the rational numbers and the real numbers have a structure. They interact in certain ways. We can look for things that are not numbers, but which follow rules like that for addition and for multiplication. Sometimes even for powers and for roots. Some of these can be strange: polynomials themselves, for example, follow rules like those of arithmetic. Matrices, which we can represent as grids of numbers, can have powers and even something like roots. Arithmetic is inspiration to finding mathematical structures that look little like our arithmetic. We can find things that follow mathematical operations but which don’t have a Fundamental Theorem of Arithmetic.

And there are more related ideas. These are often very useful. There’s modular arithmetic, in which we adjust the rules of addition and multiplication so that we can work with a finite set of numbers. There’s floating point arithmetic, in which we set machines to do our calculations. These calculations are no longer precise. But they are fast, and reliable, and that is often what we need.

So arithmetic is what people who aren’t mathematicians figure mathematicians do all day. And they are mistaken, but not by much. Arithmetic gives us an idea of what mathematics we can hope to understand. So it structures the way we think about mathematics.

## Reading the Comics, July 22, 2017: Counter-mudgeon Edition

I’m not sure there is an overarching theme to the past week’s gifts from Comic Strip Master Command. If there is, it’s that I feel like some strips are making cranky points and I want to argue against their cases. I’m not sure what the opposite of a curmudgeon is. So I shall dub myself, pending a better idea, a counter-mudgeon. This won’t last, as it’s not really a good name, but there must be a better one somewhere. We’ll see it, now that I’ve said I don’t know what it is.

Niklas Eriksson’s Carpe Diem for the 17th features the blackboard full of equations as icon for serious, deep mathematical work. It also features rabbits, although probably not for their role in shaping mathematical thinking. Rabbits and their breeding were used in the simple toy model that gave us Fibonacci numbers, famously. And the population of Arctic hares gives those of us who’ve reached differential equations a great problem to do. The ecosystem in which Arctic hares live can be modelled very simply, as hares and a generic predator. We can model how the populations of both grow with simple equations that nevertheless give us surprises. In a rich, diverse ecosystem we see a lot of population stability: one year where an animal is a little more fecund than usual doesn’t matter much. In the sparse ecosystem of the Arctic, and the one we’re building worldwide, small changes can have matter enormously. We can even produce deterministic chaos, in which if we knew exactly how many hares and predators there were, and exactly how many of them would be born and exactly how many would die, we could predict future populations. But the tiny difference between our attainable estimate and the reality, even if it’s as small as one hare too many or too few in our model, makes our predictions worthless. It’s thrilling stuff.

Vic Lee’s Pardon My Planet for the 17th reads, to me, as a word problem joke. The talk about how much change Marian should get back from Blake could be any kind of minor hassle in the real world where one friend covers the cost of something for another but expects to be repaid. But counting how many more nickels one person has than another? That’s of interest to kids and to story-problem authors. Who else worries about that count?

Jef Mallet’s Frazz for the 17th straddles that triple point joining mathematics, philosophy, and economics. It seems sensible, in an age that embraces the idea that everything can be measured, to try to quantify happiness. And it seems sensible, in age that embraces the idea that we can model and extrapolate and act on reasonable projections, to try to see what might improve our happiness. This is so even if it’s as simple as identifying what we should or shouldn’t be happy about. Caulfield is circling around the discovery of utilitarianism. It’s a philosophy that (for my money) is better-suited to problems like how ought the city arrange its bus lines than matters too integral to life. But it, too, can bring comfort.

Corey Pandolph’s Barkeater Lake rerun for the 20th features some mischievous arithmetic. I’m amused. It turns out that people do have enough of a number sense that very few people would let “17 plus 79 is 4,178” pass without comment. People might not be able to say exactly what it is, on a glance. If you answered that 17 plus 79 was 95, or 102, most people would need to stop and think about whether either was right. But they’re likely to know without thinking that it can’t be, say, 56 or 206. This, I understand, is so even for people who aren’t good at arithmetic. There is something amazing that we can do this sort of arithmetic so well, considering that there’s little obvious in the natural world that would need the human animal to add 17 and 79. There are things about how animals understand numbers which we don’t know yet.

Alex Hallatt’s Human Cull for the 21st seems almost a direct response to the Barkeater Lake rerun. Somehow “making change” is treated as the highest calling of mathematics. I suppose it has a fair claim to the title of mathematics most often done. Still, I can’t get behind Hallatt’s crankiness here, and not just because Human Cull is one of the most needlessly curmudgeonly strips I regularly read. For one, store clerks don’t need to do mathematics. The cash registers do all the mathematics that clerks might need to do, and do it very well. The machines are cheap, fast, and reliable. Not using them is an affectation. I’ll grant it gives some charm to antiques shops and boutiques where they write your receipt out by hand, but that’s for atmosphere, not reliability. And it is useful the clerk having a rough idea what the change should be. But that’s just to avoid the risk of mistakes getting through. No matter how mathematically skilled the clerk is, there’ll sometimes be a price entered wrong, or the customer’s money counted wrong, or a one-dollar bill put in the five-dollar bill’s tray, or a clerk picking up two nickels when three would have been more appropriate. We should have empathy for the people doing this work.

## Reading the Comics, June 26, 2017: Deluge Edition, Part 1

So this past week saw a lot of comic strips with some mathematical connection put forth. There were enough just for the 26th that I probably could have done an essay with exclusively those comics. So it’s another split-week edition, which suits me fine as I need to balance some of my writing loads the next couple weeks for convenience (mine).

Tony Cochrane’s Agnes for the 25th of June is fun as the comic strip almost always is. And it’s even about estimation, one of the things mathematicians do way more than non-mathematicians expect. Mathematics has a reputation for precision, when in my experience it’s much more about understanding and controlling error methods. Even in analysis, the study of why calculus works, the typical proof amounts to showing that the difference between what you want to prove and what you can prove is smaller than your tolerance for an error. So: how do we go about estimating something difficult, like, the number of stars? If it’s true that nobody really knows, how do we know there are some wrong answers? And the underlying answer is that we always know some things, and those let us rule out answers that are obviously low or obviously high. We can make progress.

Russell Myers’s Broom Hilda for the 25th is about one explanation given for why time keeps seeming to pass faster as one age. This is a mathematical explanation, built on the idea that the same linear unit of time is a greater proportion of a young person’s lifestyle so of course it seems to take longer. This is probably partly true. Most of our senses work by a sense of proportion: it’s easy to tell a one-kilogram from a two-kilogram weight by holding them, and easy to tell a five-kilogram from a ten-kilogram weight, but harder to tell a five from a six-kilogram weight.

As ever, though, I’m skeptical that anything really is that simple. My biggest doubt is that it seems to me time flies when we haven’t got stories to tell about our days, when they’re all more or less the same. When we’re doing new or exciting or unusual things we remember more of the days and more about the days. A kid has an easy time finding new things, and exciting or unusual things. Broom Hilda, at something like 1500-plus years old and really a dour, unsociable person, doesn’t do so much that isn’t just like she’s done before. Wouldn’t that be an influence? And I doubt that’s a complete explanation either. Real things are more complicated than that yet.

Mac and Bill King’s Magic In A Minute for the 25th features a form-a-square puzzle using some triangles. Mathematics? Well, logic anyway. Also a good reminder about open-mindedness when you’re attempting to construct something.

Norm Feuti’s Retail for the 26th is about how you get good at arithmetic. I suspect there’s two natural paths; you either find it really interesting in your own right, or you do it often enough you want to find ways to do it quicker. Marla shows the signs of learning to do arithmetic quickly because she does it a lot: turning “30 percent off” into “subtract ten percent three times over” is definitely the easy way to go. The alternative is multiplying by seven and dividing by ten and you don’t want to multiply by seven unless the problem gives a good reason why you should. And I certainly don’t fault the customer not knowing offhand what 30 percent off $25 would be. Why would she be in practice doing this sort of problem? Johnny Hart’s Back To B.C. for the 26th reruns the comic from the 30th of December, 1959. In it … uh … one of the cavemen guys has found his calendar for the next year has too many days. (Think about what 1960 was.) It’s a common problem. Every calendar people have developed has too few or too many days, as the Earth’s daily rotations on its axis and annual revolution around the sun aren’t perfectly synchronized. We handle this in many different ways. Some calendars worry little about tracking solar time and just follow the moon. Some calendars would run deliberately short and leave a little stretch of un-named time before the new year started; the ancient Roman calendar, before the addition of February and January, is famous in calendar-enthusiast circles for this. We’ve now settled on a calendar which will let the nominal seasons and the actual seasons drift out of synch slowly enough that periodic changes in the Earth’s orbit will dominate the problem before the error between actual-year and calendar-year length will matter. That’s a pretty good sort of error control. 8,978,432 is not anywhere near the number of days that would be taken between 4,000 BC and the present day. It’s not a joke about Bishop Ussher’s famous research into the time it would take to fit all the Biblically recorded events into history. The time is something like 24,600 years ago, a choice which intrigues me. It would make fair sense to declare, what the heck, they lived 25,000 years ago and use that as the nominal date for the comic strip. 24,600 is a weird number of years. Since it doesn’t seem to be meaningful I suppose Hart went, simply enough, with a number that was funny just for being riotously large. Mark Tatulli’s Heart of the City for the 26th places itself on my Grand Avenue warning board. There’s plenty of time for things to go a different way but right now it’s set up for a toxic little presentation of mathematics. Heart, after being grounded, was caught sneaking out to a slumber party and now her mother is sending her to two weeks of Math Camp. I’m supposing, from Tatulli’s general attitude about how stuff happens in Heart and in Lio that Math Camp will not be a horrible, penal experience. But it’s still ominous talk and I’m watching. Brian Fies’s Mom’s Cancer story for the 26th is part of the strip’s rerun on GoComics. (Many comic strips that have ended their run go into eternal loops on GoComics.) This is one of the strips with mathematical content. The spatial dimension of a thing implies relationships between the volume (area, hypervolume, whatever) of a thing and its characteristic linear measure, its diameter or radius or side length. It can be disappointing. Nicholas Gurewitch’s Perry Bible Fellowship for the 26th is a repeat of one I get on my mathematics Twitter friends now and then. Should warn, it’s kind of racy content, at least as far as my usual recommendations here go. It’s also a little baffling because while the reveal of the unclad woman is funny … what, exactly, does it mean? The symbols don’t mean anything; they’re just what fits graphically. I think the strip is getting at Dr Loring not being able to see even a woman presenting herself for sex as anything but mathematics. I guess that’s funny, but it seems like the idea isn’t quite fully developed. Zach Weinersmith’s Saturday Morning Breakfast Cereal Again for the 26th has a mathematician snort about plotting a giraffe logarithmically. This is all about representations of figures. When we plot something we usually start with a linear graph: a couple of axes perpendicular to one another. A unit of movement in the direction of any of those axes represents a constant difference in whatever that axis measures. Something growing ten units larger, say. That’s fine for many purposes. But we may want to measure something that changes by a power law, or that grows (or shrinks) exponentially. Or something that has some region where it’s small and some region where it’s huge. Then we might switch to a logarithmic plot. Here the same difference in space along the axis represents a change that’s constant in proportion: something growing ten times as large, say. The effective result is to squash a shape down, making the higher points more nearly flat. And to completely smother Weinersmith’s fine enough joke: I would call that plot semilogarithmically. I’d use a linear scale for the horizontal axis, the gazelle or giraffe head-to-tail. But I’d use a logarithmic scale for the vertical axis, ears-to-hooves. So, linear in one direction, logarithmic in the other. I’d be more inclined to use “logarithmic” plots to mean logarithms in both the horizontal and the vertical axes. Those are useful plots for turning up power laws, like the relationship between a planet’s orbital radius and the length of its year. Relationships like that turn into straight lines when both axes are logarithmically spaced. But I might also describe that as a “log-log plot” in the hopes of avoiding confusion. ## Something Cute I Never Noticed Before About Infinite Sums This is a trifle, for which I apologize. I’ve been sick. But I ran across this while reading Carl B Boyer’s The History of the Calculus and its Conceptual Development. This is from the chapter “A Century Of Anticipation”, developments leading up to Newton and Leibniz and The Calculus As We Know It. In particular, while working out the indefinite integrals for simple powers — x raised to a whole number — John Wallis, whom you’ll remember from such things as the first use of the ∞ symbol and beating up Thomas Hobbes for his lunch money, noted this: $\frac{0 + 1}{1 + 1} = \frac{1}{2}$ Which is fine enough. But then Wallis also noted that $\frac{0 + 1 + 2}{2 + 2 + 2} = \frac{1}{2}$ And furthermore that $\frac{0 + 1 + 2 + 3}{3 + 3 + 3 + 3} = \frac{1}{2}$ $\frac{0 + 1 + 2 + 3 + 4}{4 + 4 + 4 + 4 + 4} = \frac{1}{2}$ $\frac{0 + 1 + 2 + 3 + 4 + 5}{5 + 5 + 5 + 5 + 5 + 5} = \frac{1}{2}$ And isn’t that neat? Wallis goes on to conclude that this is true not just for finitely many terms in the numerator and denominator, but also if you carry on infinitely far. This seems like a dangerous leap to make, but they treated infinities and infinitesimals dangerously in those days. What makes this work is — well, it’s just true; explaining how that can be is kind of like explaining how it is circles have a center point. All right. But we can prove that this has to be true at least for finite terms. A sum like 0 + 1 + 2 + 3 is an arithmetic progression. It’s the sum of a finite number of terms, each of them an equal difference from the one before or the one after (or both). Its sum will be equal to the number of terms times the arithmetic mean of the first and last. That is, it’ll be the number of terms times the sum of the first and the last terms and divided that by two. So that takes care of the numerator. If we have the sum 0 + 1 + 2 + 3 + up to whatever number you like which we’ll call ‘N’, we know its value has to be (N + 1) times N divided by 2. That takes care of the numerator. The denominator, well, that’s (N + 1) cases of the number N being added together. Its value has to be (N + 1) times N. So the fraction is (N + 1) times N divided by 2, itself divided by (N + 1) times N. That’s got to be one-half except when N is zero. And if N were zero, well, that fraction would be 0 over 0 and we know what kind of trouble that is. It’s a tiny bit, although you can use it to make an argument about what to expect from $\int{x^n dx}$, as Wallis did. And it delighted me to see and to understand why it should be so. ## Reading the Comics, May 13, 2017: Quiet Tuesday Through Saturday Edition From the Sunday and Monday comics pages I was expecting another banner week. And then there was just nothing from Tuesday on, at least not among the comic strips I read. Maybe Comic Strip Master Command has ordered jokes saved up for the last weeks before summer vacation. Tony Cochrane’s Agnes for the 7th is a mathematics anxiety strip. It’s well-expressed, since Cochrane writes this sort of hyperbole well. It also shows a common attitude that words and stories are these warm, friendly things, while mathematics and numbers are cold and austere. Perhaps Agnes is right to say some of the problem is familiarity. It’s surely impossible to go a day without words, if you interact with people or their legacies; to go without numbers … well, properly impossible. There’s too many things that have to be counted. Or places where arithmetic sneaks in, such as getting enough money to buy a thing. But those don’t seem to be the kinds of mathematics people get anxious about. Figuring out how much change, that’s different. I suppose some of it is familiarity. It’s easier to dislike stuff you don’t do often. The unfamiliar is frightening, or at least annoying. And humans are story-oriented. Even nonfiction forms stories well. Mathematics … has stories, as do all human projects. But the mathematics itself? I don’t know. There’s just beautiful ingenuity and imagination in a lot of it. I’d just been thinking of the just beautiful scheme for calculating logarithms from a short table. But it takes time to get to that beauty. Gary Wise and Lance Aldrich’s Real Life Adventures for the 7th is a fractions joke. It might also be a joke about women concealing their ages. Or perhaps it’s about mathematicians expressing things in needlessly complicated ways. I think that’s less a mathematician’s trait than a common human trait. If you’re expert in a thing it’s hard to resist the puckish fun of showing that expertise off. Or just sowing confusion where one may. Daniel Shelton’s Ben for the 8th is a kid-doing-arithmetic problem. Even I can’t squeeze some deeper subject meaning out of it, but it’s a slow week so I’ll include the strip anyway. Sorry. Brian Boychuk and Ron Boychuk’s Chuckle Brothers for the 8th is the return of anthropomorphic-geometry joke after what feels like months without. I haven’t checked how long it’s been without but I’m assuming you’ll let me claim that. Thank you. ## Reading the Comics, April 29, 2017: The Other Half Of The Week Edition I’d been splitting Reading the Comics posts between Sunday and Thursday to better space them out. But I’ve got something prepared that I want to post Thursday, so I’ll bump this up. Also I had it ready to go anyway so don’t gain anything putting it off another two days. Bill Amend’s FoxTrot Classics for the 27th reruns the strip for the 4th of May, 2006. It’s another probability problem, in its way. Assume Jason is honest in reporting whether Paige has picked his number correctly. Assume that Jason picked a whole number. (This is, I think, the weakest assumption. I know Jason Fox’s type and he’s just the sort who’d pick an obscure transcendental number. They’re all obscure after π and e.) Assume that Jason is equally likely to pick any of the whole numbers from 1 to one billion. Then, knowing nothing about what numbers Jason is likely to pick, Paige would have one chance in a billion of picking his number too. Might as well call it certainty that she’ll pay a dollar to play the game. How much would she have to get, in case of getting the number right, to come out even or ahead? … And now we know why Paige is still getting help on probability problems in the 2017 strips. Jeff Stahler’s Moderately Confused for the 27th gives me a bit of a break by just being a snarky word problem joke. The student doesn’t even have to resist it any. Sandra Bell-Lundy’s Between Friends for the 29th also gives me a bit of a break by just being a Venn Diagram-based joke. At least it’s using the shape of a Venn Diagram to deliver the joke. It’s not really got the right content. Harley Schwadron’s 9 to 5 for the 29th is this week’s joke about arithmetic versus propaganda. It’s a joke we’re never really going to be without again. ## Reading the Comics, April 1, 2017: Connotations Edition Last week ended with another little string of mathematically-themed comic strips. Most of them invited, to me, talk about the cultural significance of mathematics and what connotations they have. So, this title for an artless essay. Berkeley Breathed’s Bloom County 2017 for the 28th of March uses “two plus two equals” as the definitive, inarguable truth. It always seems to be “two plus two”, doesn’t it? Never “two plus three”, never “three plus three”. I suppose I’ve sometimes seen “one plus one” or “two times two”. It’s easy to see why it should be a simple arithmetic problem, nothing with complicated subtraction or division or numbers as big as six. Maybe the percussive alliteration of those repeated two’s drives the phrase’s success. But then why doesn’t “two times two” show up nearly as often? Maybe the phrase isn’t iambic enough. “Two plus two” allows (to my ear) the “plus” sink in emphasis, while “times” stays a little too prominent. We need a wordsmith in to explore it. (I’m open to other hypotheses, including that “two times two” gets used more than my impression says.) Christiann MacAuley’s Sticky Comics for the 28th uses mathematics as the generic “more interesting than people” thing that nerds think about. The thing being thought of there is the Mandelbrot Set. It’s built on complex-valued numbers. Pick a complex number, any you like; that’s called ‘C’. Square the number and add ‘C’ back to itself. This will be some new complex-valued number. Square that new number and add the original ‘C’ back to it again. Square that new number and add the original ‘C’ back once more. And keep at this. There are two things that might happen. These squared numbers might keep growing infinitely large. They might be negative, or imaginary, or (most likely) complex-valued, but their size keeps growing. Or these squared numbers might not grow arbitrarily large. The Mandelbrot Set is the collection of ‘C’ values for which the numbers don’t just keep growing in size. That’s the sort of lumpy kidney bean shape with circles and lightning bolts growing off it that you saw on every pop mathematics book during the Great Fractal Boom of the 80s and 90s. There’s almost no point working it out in your head; the great stuff about fractals almost requires a computer. They take a lot of computation. But if you’re just avoiding conversation, well, anything will do. Olivia Walch’s Imogen Quest for the 29th riffs on the universe-as-simulation hypothesis. It’s one of those ideas that catches the mind and is hard to refute as long as we don’t talk to the people in the philosophy department, which we’re secretly scared of. Anyway the comic shows one of the classic uses of statistical modeling: try out a number of variations of a model in the hopes of understanding real-world behavior. This is an often-useful way to balance how the real world has stuff going on that’s important and that we don’t know about, or don’t know how to handle exactly. Mason Mastroianni’s The Wizard of Id for the 31st uses a sprawl of arithmetic as symbol of … well, of status, really. The sort of thing that marks someone a white-collar criminal. I suppose it also fits with the suggestion of magic that accompanies huge sprawls of mathematical reasoning. Bundle enough symbols together and it looks like something only the intellectual aristocracy, or at least secret cabal, could hope to read. Bob Shannon’s Tough Town for the 1st name-drops arithmetic. And shows off the attitude that anyone we find repulsive must also be stupid, as proven by their being bad at arithmetic. I admit to having no discernable feelings about the Kardashians; but I wouldn’t be so foolish as to conflate intelligence and skill-at-arithmetic. ## Reading the Comics, March 27, 2017: Not The March 26 Edition My guide for how many comics to include in one of these essays is “at least five, if possible”. Occasionally there’s a day when Comic Strip Master Command sends that many strips at once. Last Sunday was almost but not quite such a day. But the business of that day did mean I had enough strips to again divide the past week’s entries. Look for more comics in a few days, if all goes well here. Thank you. Mark Anderson’s Andertoons for the 26th reminds me of something I had wholly forgot about: decimals inside fractions. And now that this little horror’s brought back I remember my experience with it. Decimals in fractions aren’t, in meaning, any different from division of decimal numbers. And the decimals are easily enough removed. But I get the kid’s horror. Fractions and decimals are both interesting in the way they represent portions of wholes. They spend so much time standing independently of one another it feels disturbing to have them interact. Well, Andertoons kid, maybe this will comfort you: somewhere along the lines decimals in fractions just stop happening. I’m not sure when. I don’t remember when the last one passed my experience. Hector Cantu and Carlos Castellanos’s Baldo for the 26th is built on a riddle. It’s one that depends on working in shifting addition from “what everybody means by addition” to “what addition means on a clock”. You can argue — I’m sure Gracie would — that “11 plus 3” does not mean “eleven o’clock plus three hours”. But on what grounds? If it’s eleven o’clock and you know something will happen in three hours, “two o’clock” is exactly what you want. Underlying all of mathematics are definitions about what we mean by stuff like “eleven” and “plus” and “equals”. And underlying the definitions is the idea that “here is a thing we should like to know”. Addition of hours on a clock face — I never see it done with minutes or seconds — is often used as an introduction to modulo arithmetic. This is arithmetic on a subset of the whole numbers. For example, we might use 0, 1, 2, and 3. Addition starts out working the way it does in normal numbers. But then 1 + 3 we define to be 0. 2 + 3 is 1. 3 + 3 is 2. 2 + 2 is 0. 2 + 3 is 1 again. And so on. We get subtraction the same way. This sort of modulo arithmetic has practical uses. Many cryptography schemes rely on it, for example. And it has pedagogical uses; modulo arithmetic turns up all over a mathematics major’s Introduction to Not That Kind Of Algebra Course. You can use it to learn a lot of group theory with something a little less exotic than rotations and symmetries of polygonal shapes or permutations of lists of items. A clock face doesn’t quite do it, though. We have to pretend the ’12’ at the top is a ‘0’. I’ve grown more skeptical about whether appealing to clocks is useful in introducing modulo arithmetic. But it’s been a while since I’ve needed to discuss the matter at all. Rob Harrell’s Big Top rerun for the 26th mentions sudoku. Remember when sudoku was threatening to take over the world, or at least the comics page? Also, remember comics pages? Good times. It’s not one of my hobbies, but I get the appeal. Bob Shannon’s Tough Town I’m not sure if I’ve featured here before. It’s one of those high concept comics. The patrons at a bar are just what you see on the label, and there’s a lot of punning involved. Now that I’ve over-explained the joke please enjoy the joke. There are a couple of strips prior to this one featuring the same characters; they just somehow didn’t mention enough mathematics words for me to bring up here. Norm Feuti’s Retail for the 27th is about the great concern-troll of mathematics education: can our cashiers make change? I’m being snottily dismissive. Shops, banks, accountants, and tax registries are surely the most common users of mathematics — at least arithmetic — out there. And if people are going to do a thing, ordinarily, they ought to be able to do it well. But, of course, the computer does arithmetic extremely well. Far better, or at least more indefatigably, than any cashier is going to be able to do. The computer will also keep track of the prices of everything, and any applicable sales or discounts, more reliably than the mere human will. The whole point of the Industrial Revolution was to divide tasks up and assign them to parties that could do the separate parts better. Why get worked up about whether you imagine the cashier knows what$22.14 minus $16.89 is? I will say the time the bookstore where I worked lost power all afternoon and we had to do all the transactions manually we ended up with only a one-cent discrepancy in the till, thank you. ## Reading the Comics, March 25, 2017: Slow Week Edition Slow week around here for mathematically-themed comic strips. These happen. I suspect Comic Strip Master Command is warning me to stop doing two-a-week essays on reacting to comic strips and get back to more original content. Message received. If I can get ahead of some projects Monday and Tuesday we’ll get more going. Patrick Roberts’s Todd the Dinosaur for the 20th is a typical example of mathematics being something one gets in over one’s head about. Of course it’s fractions. Is there anything in elementary school that’s a clearer example of something with strange-looking rules and processes for some purpose students don’t even know what they are? In middle school and high school we get algebra. In high school there’s trigonometry. In high school and college there’s calculus. In grad school there’s grad school. There’s always something. Jeff Stahler’s Moderately Confused for the 21st is the usual bad-mathematics-of-politicians joke. It may be a little more on point considering the Future Disgraced Former President it names, but the joke is surely as old as politicians and hits all politicians with the same flimsiness. John Graziano’s Ripley’s Believe It Or Not for the 22nd names Greek mathematician Pythagoras. That’s close enough to on-point to include here, especially considering what a slow week it’s been. It may not be fair to call Pythagoras a mathematician. My understanding is we don’t know that actually did anything in mathematics, significant or otherwise. His cult attributed any of its individuals’ discoveries to him, and may have busied themselves finding other, unrelated work to credit to their founder. But there’s so much rumor and gossip about Pythagoras that it’s probably not fair to automatically dismiss any claim about him. The beans thing I don’t know about. I would be skeptical of anyone who said they were completely sure. Vic Lee’s Pardon My Planet for the 23rd is the usual sort of not-understanding-mathematics joke. In this case it’s about percentages, which are good for baffling people who otherwise have a fair grasp on fractions. I wonder if people would be better at percentages if they learned to say “percent” as “out of a hundred” instead. I’m sure everyone who teaches percentages teaches that meaning, but that doesn’t mean the warning communicates. Stephan Pastis’s Pearls Before Swine for the 24th jams a bunch of angle puns into its six panels. I think it gets most of the basic set in there. Samson’s Dark Side Of The Horse for the 25th mentions sudokus, and that’s enough for a slow week like this. I thought Horace was reaching for a calculator in the last panel myself, and was going to say that wouldn’t help any. But then I checked the numbers in the boxes and that made it all better. ## Reading the Comics, March 11, 2017: Accountants Edition And now I can wrap up last week’s delivery from Comic Strip Master Command. It’s only five strips. One certainly stars an accountant. one stars a kid that I believe is being coded to read as an accountant. The rest, I don’t know. I pick Edition titles for flimsy reasons anyway. This’ll do. Ryan North’s Dinosaur Comics for the 6th is about things that could go wrong. And every molecule of air zipping away from you at once is something which might possibly happen but which is indeed astronomically unlikely. This has been the stuff of nightmares since the late 19th century made probability an important part of physics. The chance all the air near you would zip away at once is impossibly unlikely. But such unlikely events challenge our intuitions about probability. An event that has zero chance of happening might still happen, given enough time and enough opportunities. But we’re not using our time well to worry about that. If nothing else, even if all the air around you did rush away at once, it would almost certainly rush back right away. Steve Kelley and Jeff Parker’s Dustin for the 7th of March talks about the SATs and the chance of picking right answers on a multiple-choice test. I haven’t heard about changes to the SAT but I’ll accept what the comic strip says about them for the purpose of discussion here. At least back when I took it the SAT awarded one point to the raw score for a correct answer, and subtracted one-quarter point for a wrong answer. (The raw scores were then converted into a 200-to-800 range.) I liked this. If you had no idea and guessed on answers you should expect to get one in five right and four in five wrong. On average then you would expect no net change to your raw score. If one or two wrong answers can be definitely ruled out then guessing from the remainder brings you a net positive. I suppose the change, if it is being done, is meant to be confident only right answers are rewarded. I’m not sure this is right; it seems to me there’s value in being able to identify certainly wrong answers even if the right one isn’t obvious. But it’s not my test and I don’t expect to need to take it again either. I can expression opinions without penalty. Mark Anderson’s Andertoons for the 7th is the Mark Anderson’s Andertoons for last week. It’s another kid-at-the-chalkboard panel. What gets me is that if the kid did keep one for himself then shouldn’t he have written 38? Brian Basset’s Red and Rover for the 8th mentions fractions. It’s just there as the sort of thing a kid doesn’t find all that naturally compelling. That’s all right I like the bug-eyed squirrel in the first panel. Bill Holbrook’s On The Fastrack for the 9th concludes the wedding of accountant Fi. It uses the square root symbol so as to make the cake topper clearly mathematical as opposed to just an age. ## Reading the Comics, February 3, 2017: Counting Edition And now I can close out last week’s mathematically-themed comic strips. Two of them are even about counting, which is enough for me to make that the name of this set. John Allen’s Nest Heads for the 2nd mentions a probability and statistics class and something it’s supposed to be good for. I would agree that probability and statistics are probably (I can’t find a better way to write this) the most practically useful mathematics one can learn. At least once you’re past arithmetic. They’re practical by birth; humans began studying them because they offer guidance in uncertain situations. And one can use many of their tools without needing more than arithmetic. I’m not so staunchly anti-lottery as many mathematics people are. I’ll admit I play it myself, when the jackpot is large enough. When the expectation value of the prize gets to be positive, it’s harder to rationalize not playing. This happens only once or twice a year, but it’s fun to watch and see when it happens. I grant it’s a foolish way to use two dollars (two tickets are my limit), but you know? My budget is not so tight I can’t spend four dollars foolishly a year. Besides, I don’t insist on winning one of those half-billion-dollar prizes. I imagine I’d be satisfied if I brought in a mere$10,000.

Rick Detorie’s One Big Happy for the 3rd continues my previous essay’s bit of incompetence at basic mathematics, here, counting. But working out that her age is between 22 an a gazillion may be worth doing. It’s a common mathematical challenge to find a correct number starting from little information about it. Usually we find it by locating bounds: the number must be larger than this and smaller than that. And then get the bounds closer together. Stop when they’re close enough for our needs, if we’re numerical mathematicians. Stop when the bounds are equal to each other, if we’re analytic mathematicians. That can take a lot of work. Many problems in number theory amount to “improve our estimate of the lowest (or highest) number for which this is true”. We have to start somewhere.

Samson’s Dark Side of the Horse for the 3rd is a counting-sheep joke and I was amused that the counting went so awry here. On looking over the strip again for this essay, though, I realize I read it wrong. It’s the fences that are getting counted, not the sheep. Well, it’s a cute little sheep having the same problems counting that Horace has. We don’t tend to do well counting more than around seven things at a glance. We can get a bit farther if we can group things together and spot that, say, we have four groups of four fences each. That works and it’s legitimate; we’re counting and we get the right count out of it. But it does feel like we’re doing something different from how we count, say, three things at a glance.

Mick Mastroianni and Mason MastroianniDogs of C Kennel for the 3rd is about the world’s favorite piece of statistical mechanics, entropy. There’s room for quibbling about what exactly we mean by thermodynamics saying all matter is slowly breaking down. But the gist is fair enough. It’s still mysterious, though. To say that the disorder of things is always increasing forces us to think about what we mean by disorder. It’s easy to think we have an idea what we mean by it. It’s hard to make that a completely satisfying definition. In this way it’s much like randomness, which is another idea often treated as the same as disorder.

Bill Amend’s FoxTrot Classics for the 3rd reprinted the comic from the 10th of February, 2006. Mathematics teachers always want to see how you get your answers. Why? … Well, there are different categories of mistakes someone can make. One can set out trying to solve the wrong problem. One can set out trying to solve the right problem in a wrong way. One can set out solving the right problem in the right way and get lost somewhere in the process. Or one can be doing just fine and somewhere along the line change an addition to a subtraction and get what looks like the wrong answer. Each of these is a different kind of mistake. Knowing what kinds of mistakes people make is key to helping them not make these mistakes. They can get on to making more exciting mistakes.

## Reading the Comics, February 2, 2017: I Haven’t Got A Jumble Replacement Source Yet

If there was one major theme for this week it was my confidence that there must be another source of Jumble strips out there. I haven’t found it, but I admit not making it a priority either. The official Jumble site says I can play if I activate Flash, but I don’t have enough days in the year to keep up with Flash updates. And that doesn’t help me posting mathematics-relevant puzzles here anyway.

Mark Anderson’s Andertoons for January 29th satisfies my Andertoons need for this week. And it name-drops the one bit of geometry everyone remembers. To be dour and humorless about it, though, I don’t think one could likely apply the Pythagorean Theorem. Typically the horizontal axis and the vertical axis in a graph like this measure different things. Squaring the different kinds of quantities and adding them together wouldn’t mean anything intelligible. What would even be the square root of (say) a squared-dollars-plus-squared-weeks? This is something one learns from dimensional analysis, a corner of mathematics I’ve thought about writing about some. I admit this particular insight isn’t deep, but everything starts somewhere.

Norm Feuti’s Gil rerun for the 30th is a geometry name-drop, listing it as the sort of category Jeopardy! features. Gil shouldn’t quit so soon. The responses for the category are “What is the Pythagorean Theorem?”, “What is acute?”, “What is parallel?”, “What is 180 degrees?” (or, possibly, 360 or 90 degrees), and “What is a pentagon?”.

Terri Libenson’s Pajama Diaries for the 1st of February shows off the other major theme of this past week, which was busy enough that I have to again split the comics post into two pieces. That theme is people getting basic mathematics wrong. Mostly counting. (You’ll see.) I know there’s no controlling what people feel embarrassed about. But I think it’s unfair to conclude you “can no longer” do mathematics in your head because you’re not able to make change right away. It’s normal to be slow or unreliable about something you don’t do often. Inexperience and inability are not the same thing, and it’s unfair to people to conflate them.

Gordon Bess’s Redeye for the 21st of September, 1970, got rerun the 1st of February. And it’s another in the theme of people getting basic mathematics wrong. And even more basic mathematics this time. There’s more problems-with-counting comics coming when I finish the comics from the past week.

Dave Whamond’s Reality Check for the 1st hopes that you won’t notice the label on the door is painted backwards. Just saying. It’s an easy joke to make about algebra, also, that it should put letters in to perfectly good mathematics. Letters are used for good reasons, though. We’ve always wanted to work out the value of numbers we only know descriptions of. But it’s way too wordy to use the whole description of the number every time we might speak of it. Before we started using letters we could use placeholder names like “re”, meaning “thing” (as in “thing we want to calculate”). That works fine, although it crashes horribly when we want to track two or three things at once. It’s hard to find words that are decently noncommittal about their values but that we aren’t going to confuse with each other.

So the alphabet works great for this. An individual letter doesn’t suggest any particular number, as long as we pretend ‘O’ and ‘I’ and ‘l’ don’t look like they do. But we also haven’t got any problem telling ‘x’ from ‘y’ unless our handwriting is bad. They’re quick to write and to say aloud, and they don’t require learning to write any new symbols.

Later, yes, letters do start picking up connotations. And sometimes we need more letters than the Roman alphabet allows. So we import from the Greek alphabet the letters that look different from their Roman analogues. That’s a bit exotic. But at least in a Western-European-based culture they aren’t completely novel. Mathematicians aren’t really trying to make this hard because, after all, they’re the ones who have to deal with the hard parts.

Bu Fisher’s Mutt and Jeff rerun for the 2nd is another of the basic-mathematics-wrong jokes. But it does get there by throwing out a baffling set of story-problem-starter points. Particularly interesting to me is Jeff’s protest in the first panel that they couldn’t have been doing 60 miles an hour as they hadn’t been out an hour. It’s the sort of protest easy to use as introduction to the ideas of average speed and instantaneous speed and, from that, derivatives.

## Reading the Comics, January 28, 2017: Chuckle Brothers Edition

The week started out quite busy and I was expecting I’d have to split my essay again. It didn’t turn out that way; Comic Strip Master Command called a big break on mathematically-themed comics from Tuesday on. And then nobody from Comics Kingdom or from Creators.com needed inclusion either. I just have a bunch of GoComics links and a heap of text here. I bet that changes by next week. Still no new Jumble strips.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 22nd was their first anthropomorphic numerals joke of the week.

Kevin Fagan’s Drabble for the 22nd uses arithmetic as the sort of problem it’s easy to get clearly right or clearly wrong. It’s a more economical use of space than (say) knowing how many moons Saturn’s known to have. (More than we thought there were as long ago as Thursday.) I do like that there’s a decent moral to this on the way to the punch line.

Bill Amend’s FoxTrot for the 22nd has Jason stand up for “torus” as a better name for doughnuts. You know how nerdy people will like putting a complicated word onto an ordinary thing. But there are always complications. A torus ordinarily describes the shape made by rotating a circle around an axis that’s in the plane of the circle. The result is a surface, though, the shell of a doughnut and none of the interior. If we’re being fussy. I don’t know of a particular name for the torus with its interior and suspect that, if pressed, a mathematician would just say “torus” or maybe “doughnut”.

We can talk about toruses in two dimensions; those look just like circles. The doughnut-shell shape is a torus in three dimensions. There’s torus shapes made by rotating spheres, or hyperspheres, in four or more dimensions. I’m not going to draw them. And we can also talk about toruses by the number of holes that go through them. If a normal torus is the shape of a ring-shaped pool toy, a double torus is the shape of a two-seater pool toy, a triple torus something I don’t imagine exists in the real world. A quadruple torus could look, I imagine, like some pool toys Roller Coaster Tycoon allows in its water parks. I’m saying nothing about whether they’re edible.

Brian Boychuk and Ron Boychuk’s The Chuckle Brothers for the 23rd was their second anthropomorphic numerals joke of the week. I suppose sometimes you just get an idea going.

Mikael Wulff and Anders Morgenthaler’s TruthFacts for the 23rd jokes about mathematics skills versus life. The growth is fine enough; after all, most of us are at, or get to, our best at something while we’re training in it or making regular use of it. So the joke peters out into the usual “I never use mathematics in real life” crack, which, eh. I agree it’s what I feel like my mathematics skills have done ever since I got my degree, at any rate.

Teresa Burritt’s Frog Applause for the 24th describes an extreme condition which hasn’t been a problem for me. I’m not an overindulgey type.

Randy Glasbergen’s Glasbergen Cartoons rerun for the 26th is the pie chart joke for this week.

Michael Fry’s Committed rerun for the 28th just riffs on the escalation of hyperbole, and what sure looks like an exponential growth of hyperbolic numbers. There’s a bit of scientific notation in the last panel. The “1 x” part isn’t necessary. It doesn’t change the value of the expression “1 x 1026”. But it might be convenient to use the “1 x” anyway. Scientific notation is about separating the size of the number from the interesting digits that the number has. Often when you compare numbers you’re interested in the size or else you’re interested in the important digits. Get into that habit and it’s not worth making an exception just because the interesting digits turn out to be boring in this case.

## Reading the Comics, January 14, 2017: Redeye and Reruns Edition

So for all I worried about the Gocomics.com redesign it’s not bad. The biggest change is it’s removed a side panel and given the space over to the comics. And while it does show comics you haven’t been reading, it only shows one per day. One week in it apparently sticks with the same comic unless you choose to dismiss that. So I’ve had it showing me The Comic Strip That Has A Finale Every Day as a strip I’m not “reading”. I’m delighted how thisbreaks the logic about what it means to “not read” an “ongoing comic strip”. (That strip was a Super-Fun-Pak Comix offering, as part of Ruben Bolling’s Tom the Dancing Bug. It was turned into a regular Gocomics.com feature by someone who got the joke.)

Comic Strip Master Command responded to the change by sending out a lot of comic strips. I’m going to have to divide this week’s entry into two pieces. There’s not deep things to say about most of these comics, but I’ll make do, surely.

Julie Larson’s Dinette Set rerun for the 8th is about one of the great uses of combinatorics. That use is working out how the number of possible things compares to the number of things there are. What’s always staggering is that the number of possible things grows so very very fast. Here one of Larson’s characters claims a science-type show made an assertion about the number of possible ideas a brain could hold. I don’t know if that’s inspired by some actual bit of pop science. I can imagine someone trying to estimate the number of possible states a brain might have.

And that has to be larger than the number of atoms in the universe. Consider: there’s something less than a googol of atoms in the universe. But a person can certainly have the idea of the number 1, or the idea of the number 2, or the idea of the number 3, or so on. I admit a certain sameness seems to exist between the ideas of the numbers 2,038,412,562,593,604 and 2,038,412,582,593,604. But there is a difference. We can out-number the atoms in the universe even before we consider ideas like rabbits or liberal democracy or jellybeans or board games. The universe never had a chance.

Or did it? Is it possible for a number to be too big for the human brain to ponder? If there are more digits in the number than there are atoms in the universe we can’t form any discrete representation of it, after all. … Except that we kind of can. For example, “the largest prime number less than one googolplex” is perfectly understandable. We can’t write it out in digits, I think. But you now have thought of that number, and while you may not know what its millionth decimal digit is, you also have no reason to care what that digit is. This is stepping into the troubled waters of algorithmic complexity.

Bob Weber Jr’s Slylock Fox and Comics for Kids for the 9th is built on soap bubbles. The link between the wand and the soap bubble vanishes quickly once the bubble breaks loose of the wand. But soap films that keep adhered to the wand or mesh can be quite strangely shaped. Soap films are a practical example of a kind of partial differential equations problem. Partial differential equations often appear when we want to talk about shapes and surfaces and materials that tug or deform the material near them. The shape of a soap bubble will be the one that minimizes the torsion stresses of the bubble’s surface. It’s a challenge to solve analytically. It’s still a good challenge to solve numerically. But you can do that most wonderful of things and solve a differential equation experimentally, if you must. It’s old-fashioned. The computer tools to do this have gotten so common it’s hard to justify going to the engineering lab and getting soapy water all over a mathematician’s fingers. But the option is there.

Gordon Bess’s Redeye rerun from the 28th of August, 1970, is one of a string of confused-student jokes. (The strip had a Generic Comedic Western Indian setting, putting it in the vein of Hagar the Horrible and other comic-anachronism comics.) But I wonder if there are kids baffled by numbers getting made several different ways. Experience with recipes and assembly instructions and the like might train someone to thinking there’s one correct way to make something. That could build a bad intuition about what additions can work.

Corey Pandolph’s Barkeater Lake rerun for the 9th just name-drops algebra. And that as a word that starts with the “alj” sound. So far as I’m aware there’s not a clear etymological link between Algeria and algebra, despite both being modified Arabic words. Algebra comes from “al-jabr”, about reuniting broken things. Algeria comes from Algiers, which Wikipedia says derives from `al-jaza’ir”, “the Islands [of the Mazghanna tribe]”.

Guy Gilchrist’s Nancy for the 9th is another mathematics-cameo strip. But it was also the first strip I ran across this week that mentioned mathematics and wasn’t a rerun. I’ll take it.

Donna A Lewis’s Reply All for the 9th has Lizzie accuse her boyfriend of cheating by using mathematics in Scrabble. He seems to just be counting tiles, though. I think Lizzie suspects something like Blackjack card-counting is going on. Since there are only so many of each letter available knowing just how many tiles remain could maybe offer some guidance how to play? But I don’t see how. In Blackjack a player gets to decide whether to take more cards or not. Counting cards can suggest whether it’s more likely or less likely that another card will make the player or dealer bust. Scrabble doesn’t offer that choice. One has to refill up to seven tiles until the tile bag hasn’t got enough left. Perhaps I’m overlooking something; I haven’t played much Scrabble since I was a kid.

Perhaps we can take the strip as portraying the folk belief that mathematicians get to know secret, barely-explainable advantages on ordinary folks. That itself reflects a folk belief that experts of any kind are endowed with vaguely cheating knowledge. I’ll admit being able to go up to a blackboard and write with confidence a bunch of integrals feels a bit like magic. This doesn’t help with Scrabble.

Gordon Bess’s Redeye continued the confused-student thread on the 29th of August, 1970. This one’s a much older joke about resisting word problems.

Ryan North’s Dinosaur Comics rerun for the 10th talks about multiverses. If we allow there to be infinitely many possible universes that would suggest infinitely many different Shakespeares writing enormously many variations of everything. It’s an interesting variant on the monkeys-at-typewriters problem. I noticed how T-Rex put Shakespeare at typewriters too. That’ll have many of the same practical problems as monkeys-at-typewriters do, though. There’ll be a lot of variations that are just a few words or a trivial scene different from what we have, for example. Or there’ll be variants that are completely uninteresting, or so different we can barely recognize them as relevant. And that’s if it’s actually possible for there to be an alternate universe with Shakespeare writing his plays differently. That seems like it should be possible, but we lack evidence that it is.

## The End 2016 Mathematics A To Z: Quotient Groups

I’ve got another request today, from the ever-interested and group-theory-minded gaurish. It’s another inspirational one.

## Quotient Groups.

We all know about even and odd numbers. We don’t have to think about them. That’s why it’s worth discussing them some.

We do know what they are, though. The integers — whole numbers, positive and negative — we can split into two sets. One of them is the even numbers, two and four and eight and twelve. Zero, negative two, negative six, negative 2,038. The other is the odd numbers, one and three and nine. Negative five, negative nine, negative one.

What do we know about numbers, if all we look at is whether numbers are even or odd? Well, we know every integer is either an odd or an even number. It’s not both; it’s not neither.

We know that if we start with an even number, its negative is also an even number. If we start with an odd number, its negative is also an odd number.

We know that if we start with a number, even or odd, and add to it its negative then we get an even number. A specific number, too: zero. And that zero is interesting because any number plus zero is that same original number.

We know we can add odds or evens together. An even number plus an even number will be an even number. An odd number plus an odd number is an even number. An odd number plus an even number is an odd number. And subtraction is the same as addition, by these lights. One number minus an other number is just one number plus negative the other number. So even minus even is even. Odd minus odd is even. Odd minus even is odd.

We can pluck out some of the even and odd numbers as representative of these sets. We don’t want to deal with big numbers, nor do we want to deal with negative numbers if we don’t have to. So take ‘0’ as representative of the even numbers. ‘1’ as representative of the odd numbers. 0 + 0 is 0. 0 + 1 is 1. 1 + 0 is 1. The addition is the same thing we would do with the original set of integers. 1 + 1 would be 2, which is one of the even numbers, which we represent with 0. So 1 + 1 is 0. If we’ve picked out just these two numbers each is the minus of itself: 0 – 0 is 0 + 0. 1 – 1 is 1 + 1. All that gives us 0, like we should expect.

Two paragraphs back I said something that’s obvious, but deserves attention anyway. An even plus an even is an even number. You can’t get an odd number out of it. An odd plus an odd is an even number. You can’t get an odd number out of it. There’s something fundamentally different between the even and the odd numbers.

And now, kindly reader, you’ve learned quotient groups.

OK, I’ll do some backfilling. It starts with groups. A group is the most skeletal cartoon of arithmetic. It’s a set of things and some operation that works like addition. The thing-like-addition has to work on pairs of things in your set, and it has to give something else in the set. There has to be a zero, something you can add to anything without changing it. We call that the identity, or the additive identity, because it doesn’t change something else’s identity. It makes sense if you don’t stare at it too hard. Everything has an additive inverse. That is everything has a “minus”, that you can add to it to get zero.

With odd and even numbers the set of things is the integers. The thing-like-addition is, well, addition. I said groups were based on how normal arithmetic works, right?

And then you need a subgroup. A subgroup is … well, it’s a subset of the original group that’s itself a group. It has to use the same addition the original group does. The even numbers are such a subgroup of the integers. Formally they make something called a “normal subgroup”, which is a little too much for me to explain right now. If your addition works like it does for normal numbers, that is, “a + b” is the same thing as “b + a”, then all your subgroups are normal groups. Yes, it can happen that they’re not. If the addition is something like rotations in three-dimensional space, or swapping the order of things, then the order you “add” things in matters.

We make a quotient group by … OK, this isn’t going to sound like anything. It’s a group, though, like the name says. It uses the same addition that the original group does. Its set, though, that’s itself made up of sets. One of the sets is the normal subgroup. That’s the easy part.

Then there’s something called cosets. You make a coset by picking something from the original group and adding it to everything in the subgroup. If the thing you pick was from the original subgroup that’s just going to be the subgroup again. If you pick something outside the original subgroup then you’ll get some other set.

Starting from the subgroup of even numbers there’s not a lot to do. You can get the even numbers and you get the odd numbers. Doesn’t seem like much. We can do otherwise though. Suppose we start from the subgroup of numbers divisible by 4, though. That’s 0, 4, 8, 12, -4, -8, -12, and so on. Now there’s three cosets we can make from that. We can start with the original set of numbers. Or we have 1 plus that set: 1, 5, 9, 13, -3, -7, -11, and so on. Or we have 2 plus that set: 2, 6, 10, 14, -2, -6, -10, and so on. Or we have 3 plus that set: 3, 7, 11, 15, -1, -5, -9, and so on. None of these others are subgroups, which is why we don’t call them subgroups. We call them cosets.

These collections of cosets, though, they’re the pieces of a new group. The quotient group. One of them, the normal subgroup you started with, is the identity, the thing that’s as good as zero. And you can “add” the cosets together, in just the same way you can add “odd plus odd” or “odd plus even” or “even plus even”.

For example. Let me start with the numbers divisible by 4. I will have so much a better time if I give this a name. I’ll pick ‘Q’. This is because, you know, quarters, quartet, quadrilateral, this all sounds like four-y stuff. The integers — the integers have a couple of names. ‘I’, ‘J’, and ‘Z’ are the most common ones. We get ‘Z’ from German; a lot of important group theory was done by German-speaking mathematicians. I’m used to it so I’ll stick with that. The quotient group ‘Z / Q’, read “Z modulo Q”, has (it happens) four cosets. One of them is Q. One of them is “1 + Q”, that set 1, 5, 9, and so on. Another of them is “2 + Q”, that set 2, 6, 10, and so on. And the last is “3 + Q”, that set 3, 7, 11, and so on.

And you can add them together. 1 + Q plus 1 + Q turns out to be 2 + Q. Try it out, you’ll see. 1 + Q plus 2 + Q turns out to be 3 + Q. 2 + Q plus 2 + Q is Q again.

The quotient group uses the same addition as the original group. But it doesn’t add together elements of the original group, or even of the normal subgroup. It adds together sets made from the normal subgroup. We’ll denote them using some form that looks like “a + N”, or maybe “a N”, if ‘N’ was the normal subgroup and ‘a’ something that wasn’t in it. (Sometimes it’s more convenient writing the group operation like it was multiplication, because we do that by not writing anything at all, which saves us from writing stuff.)

If we’re comfortable with the idea that “odd plus odd is even” and “even plus odd is odd” then we should be comfortable with adding together quotient groups. We’re not, not without practice, but that’s all right. In the Introduction To Not That Kind Of Algebra course mathematics majors take they get a lot of practice, just in time to be thrown into rings.

Quotient groups land on the mathematics major as a baffling thing. They don’t actually turn up things from the original group. And they lead into important theorems. But to an undergraduate they all look like text huddling up to ladders of quotient groups. We’re told these are important theorems and they are. They also go along with beautiful diagrams of how these quotient groups relate to each other. But they’re hard going. It’s tough finding good examples and almost impossible to explain what a question is. It comes as a relief to be thrown into rings. By the time we come back around to quotient groups we’ve usually had enough time to get used to the idea that they don’t seem so hard.

Really, looking at odds and evens, they shouldn’t be so hard.

## Reading the Comics, November 12, 2016: Frazz and Monkeys Edition

Two things made repeat appearances in the mathematically-themed comics this week. They’re the comic strip Frazz and the idea of having infinitely many monkeys typing. Well, silly answers to word problems also turned up, but that’s hard to say many different things about. Here’s what I make the week in comics out to be.

Sandra Bell-Lundy’s Between Friends for the 6th introduces the infinite monkeys problem. I wonder sometimes why the monkeys-on-typewriters thing has so caught the public imagination. And then I remember it encourages us to stare directly into infinity and its intuition-destroying nature from the comfortable furniture of the mundane — typewriters, or keyboards, for goodness’ sake — with that childish comic dose of monkeys. Given that it’s a wonder we ever talk about anything else, really.

Monkeys writing Shakespeare has for over a century stood as a marker for what’s possible but incredibly improbable. I haven’t seen it compared to finding a four-digit PIN. It has got me wondering about the chance that four randomly picked letters will be a legitimate English word. I’m sure the chance is more than the one-in-a-thousand chance someone would guess a randomly drawn PIN correctly on one try. More than one in a hundred? I’m less sure. The easy-to-imagine thing to do is set a computer to try out all 456,976 possible sets of four letters and check them against a dictionary. The number of hits divided by the number of possibilities would be the chance of drawing a legitimate word. If I had a less capable computer, or were checking even longer words, I might instead draw some set number of words, never minding that I didn’t get every possibility. The fraction of successful words in my sample would be something close to the chance of drawing any legitimate word.

If I thought a little deeper about the problem, though, I’d just count how many four-letter words are already in my dictionary and divide that into 456,976. It’s always a mistake to start programming before you’ve thought the problem out. The trouble is not being able to tell when that thinking-out is done.

Richard Thompson’s Poor Richard’s Almanac for the 7th is the other comic strip to mention infinite monkeys. Well, chimpanzees in this case. But for the mathematical problem they’re not different. I’ve featured this particular strip before. But I’m a Thompson fan. And goodness but look at the face on the T S Eliot fan in the lower left corner there.

Jeff Mallet’s Frazz for the 6th gives Caulfield one of those flashes of insight that seems like it should be something but doesn’t mean much. He’s had several of these lately, as mentioned here last week. As before this is a fun discovery about Roman Numerals, but it doesn’t seem like it leads to much. Perhaps a discussion of how the subtractive principle — that you can write “four” as “IV” instead of “IIII” — evolved over time. But then there isn’t much point to learning Roman Numerals at all. It’s got some value in showing how much mathematics depends on culture. Not just that stuff can be expressed in different ways, but that those different expressions make different things easier or harder to do. But I suspect that isn’t the objective of lessons about Roman Numerals.

Frazz got my attention again the 12th. This time it just uses arithmetic, and a real bear of an arithmetic problem, as signifier for “a big pile of hard work”. This particular problem would be — well, I have to call it tedious, rather than hard. doing it is just a long string of adding together two numbers. But to do that over and over, by my count, at least 47 times for this one problem? Hardly any point to doing that much for one result.

Patrick Roberts’s Todd the Dinosaur for the 7th calls out fractions, and arithmetic generally, as the stuff that ruins a child’s dreams. (Well, a dinosaur child’s dreams.) Still, it’s nice to see someone reminding mathematicians that a lot of their field is mostly used by accountants. Actuaries we know about; mathematics departments like to point out that majors can get jobs as actuaries. I don’t know of anyone I went to school with who chose to become one or expressed a desire to be an actuary. But I admit not asking either.

Mike Thompson’s Grand Avenue started off a week of students-resisting-the-test-question jokes on the 7th. Most of them are hoary old word problem jokes. But, hey, I signed up to talk about it when a comic strip touches a mathematics topic and word problems do count.

Zach Weinersmith’s Saturday Morning Breakfast Cereal reprinted the 7th is a higher level of mathematical joke. It’s from the genre of nonsense calculation. This one starts off with what’s almost a cliche, at least for mathematics and physics majors. The equation it starts with, $e^{i Pi} = -1$, is true. And famous. It should be. It links exponentiation, imaginary numbers, π, and negative numbers. Nobody would have seen it coming. And from there is the sort of typical gibberish reasoning, like writing “Pi” instead of π so that it can be thought of as “P times i”, to draw to the silly conclusion that P = 0. That much work is legitimate.

From there it sidelines into “P = NP”, which is another equation famous to mathematicians and computer scientists. It’s a shorthand expression of a problem about how long it takes to find solutions. That is, how many steps it takes. How much time it would take a computer to solve a problem. You can see why it’s important to have some study of how long it takes to do a problem. It would be poor form to tie up your computer on a problem that won’t be finished before the computer dies of old age. Or just take too long to be practical.

Most problems have some sense of size. You can look for a solution in a small problem or in a big one. You expect searching for the solution in a big problem to take longer. The question is how much longer? Some methods of solving problems take a length of time that grows only slowly as the size of the problem grows. Some take a length of time that grows crazy fast as the size of the problem grows. And there are different kinds of time growth. One kind is called Polynomial, because everything is polynomials. But there’s a polynomial in the problem’s size that describes how long it takes to solve. We call this kind of problem P. Another is called Non-Deterministic Polynomial, for problems that … can’t. We assume. We don’t know. But we know some problems that look like they should be NP (“NP Complete”, to be exact).

It’s an open question whether P and NP are the same thing. It’s possible that everything we think might be NP actually can be solved by a P-class algorithm we just haven’t thought of yet. It would be a revolution in our understanding of how to find solutions if it were. Most people who study algorithms think P is not NP. But that’s mostly (as I understand it) because it seems like if P were NP then we’d have some leads on proving that by now. You see how this falls short of being rigorous. But it is part of expertise to get a feel for what seems to make sense in light of everything else we know. We may be surprised. But it would be inhuman not to have any expectations of a problem like this.

Mark Anderson’s Andertoons for the 8th gives us the Andertoons content for the week. It’s a fair question why a right triangle might have three sides, three angles, three vertices, and just the one hypotenuse. The word’s origin, from Greek, meaning “stretching under” or “stretching between”. It’s unobjectionable that we might say this is the stretch from one leg of the right triangle to another. But that leaves unanswered why there’s just the one hypothenuse, since the other two legs also stretch from the end of one leg to another. Dr Sarah on The Math Forum suggests we need to think of circles. Draw a circle and a diameter line on it. Now pick any point on the circle other than where the diameter cuts it. Draw a line from one end of the diameter to your point. And from your point to the other end of the diameter. You have a right triangle! And the hypothenuse is the leg stretching under the other two. Yes, I’m assuming you picked a point above the diameter. You did, though, didn’t you? Humans do that sort of thing.

I don’t know if Dr Sarah’s explanation is right. It sounds plausible and sensible. But those are weak pins to hang an etymology on. But I have no reason to think she’s mistaken. And the explanation might help people accept there is the one hypothenuse and there’s something interesting about it.

The first (and as I write this only) commenter, Kristiaan, has a good if cheap joke there.

## Reading the Comics, October 22, 2016: The Jokes You Can Make About Fractions Edition

Last week had a whole bundle and a half of mathematically-themed comics so let me finish off the set. Also let me refresh my appeal for words for my End Of 2016 Mathematics A To Z. There’s all sorts of letters not yet claimed; please think of a mathematical term and request it!

David L Hoyt and Jeff Knurek’s Jumble for the 19th gives us a chance to do some word puzzle games again. If you like getting the big answer without doing the individual words then pay attention to the blackboard in the comic. Just saying.

Patrick J Marran’s Francis for the 20th features origami, as well as some of the more famous polyhedrons. The study of what shapes you can make from a flat sheet by origami processes — just folding, no cutting — is a neat one. Apparently origami geometry can be built out of seven axioms. I’m delighted to learn that the axioms were laid out as recently as 1992, with the exception of one that went unnoticed until 2002.

Gabby describes her shape as an isocahedron, which must be a typo. We all make them. There’s icosahedrons which look like that figure and I’ve certainly slipped consonants around that way.

I’m surprised and delighted to find there are ways to make an origami icosahedron. Her figure doesn’t look much like the origami icosahedron of those instructions, but there are many icosahedrons. The name just means there are 20 faces to the polyhedron so there’s a lot of room for variants.

If you were wondering, yes, the Francis of the title is meant to be the Pope. It’s kind of a Pope Francis fan comic. I cannot explain this phenomenon.

Rick Detorie’s One Big Happy rerun for the 21st retells one of the standard jokes you can always make about fractions. Fortunately it uses that only as part of the setup, which shows off why I’ve long liked Detorie’s work. Good cartoonists — good writers — take a stock joke and add something to make it fit their characters.

I’ve featured Richard Thompson’s Poor Richard’s Almanac rerun from the 21st before. I’ll surely feature it again. I just like Richard Thompson art like this. This is my dubious inclusion of the essay. In “What’s New At The Zoo” he tosses off a mention of chimpanzees now typing at 120 words per minute. A comic reference to the famous thought experiment of a monkey, or a hundred monkeys, or infinitely many monkeys given typewriters and time to write all the works of literature? Maybe. Or it might just be that it’s a funny idea. It is, of course.

In Rick Kirkman and Jerry Scott’s Baby Blues for the 22nd Hammie offers multiple answers to each mathematics problem. “I like to increase my odds,” he says. For arithmetic problems, that’s not really helping. But it is often useful, especially in modeling complicated systems, to work out multiple answers. If you’re not sure how something should behave, and it’s troublesome to run experiments, then try develop several different models. If the models all describe similar behavior, then, good! It’s reason to believe you’re probably right, or at least close to right. If the models disagree about their conclusions then you need information. You need experimental results. The ways your models disagree can inspire new experiments.

Mark Leiknes’s Cow and Boy rerun for the 22nd is another with one of the standard jokes you can make about fractions. I suspect I’ve featured this before too, but I quite like Cow and Boy. It’s sad that the strip was cancelled, and couldn’t make a go of it as web comic. I’m not surprised; the strip had so many running jokes it might as well have had a deer and an orca shooting rocket-propelled grenades at new readers. But it’s grand seeing the many, many, many running jokes as they were first established. This is part of the sequence in which Billy, the Boy of the title, discovers there’s another kid named Billy in the class, quickly dubbed Smart Billy for reasons the strip makes clear.

## Reading the Comics, October 1, 2016: Jumble Is Back Edition

Comic Strip Master Command sent another normal-style week for mathematics references. There’s not much that lets me get really chatty or gossippy about mathematics lore. That’s all right. The important thing is: we’ve got Jumble back.

Greg Cravens’s The Buckets for the 25th features a bit of parental nonsense-telling. The rather annoying noise inside a car’s cabin when there’s one window open is the sort of thing fluid mechanics ought to be able to study. I see references claiming this noise to be a Helmholz Resonance. This is a kind of oscillation in the air that comes from wind blowing across the lone hole in a solid object. Wikipedia says it’s even the same phenomenon producing an ocean-roar in a seashell held up to the ear. It’s named for Hermann von Helmholtz, who described it while studying sound and vortices. Helmholz is also renowned for making a clear statement of the conservation of energy — an idea many were working towards, mind — and in thermodynamics and electromagnetism and for that matter how the eye works. Also how fast nerves transmit signals. All that said, I’m not sure that all the unpleasant sound heard and pressure felt from a single opened car window is Helmholz Resonance. Real stuff is complicated and the full story is always more complicated than that. I wouldn’t go farther than saying that Helmholz Resonance is one thing to look at.

Michael Cavna’s Warped for the 25th uses two mathematics-cliché equations as “amazingly successful formulas”. One can quibble with whether Einstein should be counted under mathematics. Pythagoras, at least for the famous theorem named for him, nobody would argue. John Grisham, I don’t know, the joke seems dated to me but we are talking about the comics.

Tony Carrillos’ F Minus for the 28th uses arithmetic as as something no reasonable person can claim is incorrect. I haven’t read the comments, but I am slightly curious whether someone says something snarky about Common Core mathematics — or even the New Math for crying out loud — before or after someone finds a base other than ten that makes the symbols correct.

Cory Thomas’s college-set soap-opera strip Watch Your Head for the 28th name-drops Introduction to Functional Analysis. It won’t surprise you it’s a class nobody would take on impulse. It’s an upper-level undergraduate or a grad-student course, something only mathematics majors would find interesting. But it is very interesting. It’s the reward students have for making it through Real Analysis, the spirit-crushing course about why calculus works. Functional Analysis is about what we can do with functions. We can make them work like numbers. We can define addition and multiplication, we can measure their size, we can create sequences of them. We can treat functions almost as if they were numbers. And while we’re working on things more abstract and more exotic than the ordinary numbers Real Analysis depends on, somehow, Functional Analysis is easier than Real Analysis. It’s a wonder.

Mark Anderson’s Andertoons for the 29th features a student getting worried about the order of arithmetic operations. I appreciate how kids get worried about the feelings of things like that. Although, truly, subtraction doesn’t go “last”; addition and subtraction have the same priority. They share the bottom of the pile, though. Multiplication and division similarly share a priority, above addition-and-subtraction. Many guides to the order of operations say to do addition-and-subtraction in order left to right, but that’s not so. Setting a left-to-right order is okay for deciding where to start. But you could do a string of additions or subtractions in any order and get the same answer, unless the expression is inconsistent.

Daniel Beyer’s Long Story Short for the 30th is a pie chart joke. There’s not a lot of mathematics to it, but I’m amused.

Justin Boyd’s Invisible Bread for the 30th has maybe my favorite dumb joke of the week. It’s just a kite that’s proven its knowledge of mathematics. I’m a little surprised the kite didn’t call out a funnier number, by which I mean 37, but perhaps … no, that doesn’t work, actually. Of course the kite would be comfortable with higher mathematics.

And as promised, David L Hoyt and Jeff Knurek’s Jumble for the 1st of October mentions mathematics. That’s enough for me to include here.

As though to reinforce how nothing was basically wrong, Comic Strip Master Command sent a normal number of mathematically themed comics around this past week. They bunched the strips up in the first half of the week, but that will happen. It was a fun set of strips in any event.

Rob Harrell’s Adam @ Home for the 11th tells of a teacher explaining division through violent means. I’m all for visualization tools and if we are going to use them, the more dramatic the better. But I suspect Mrs Clark’s students will end up confused about what exactly they’ve learned. If a doll is torn into five parts, is that communicating that one divided by five is five? If the students were supposed to identify the mass of the parts of the torn-up dolls as the result of dividing one by five, was that made clear to them? Maybe it was. But there’s always the risk in a dramatic presentation that the audience will misunderstand the point. The showier the drama the greater the risk, it seems to me. But I did only get the demonstration secondhand; who knows how well it was done?

Greg Cravens’ The Buckets for the 11th has the kid, Toby, struggling to turn a shirt backwards and inside-out without taking it off. As the commenters note this is the sort of problem we get into all the time in topology. The field is about what can we say about shapes when we don’t worry about distance? If all we know about a shape is the ways it’s connected, the number of holes it has, whether we can distinguish one side from another, what else can we conclude? I believe Gocomics.com commenter Mike is right: take one hand out the bottom of the shirt and slide it into the other sleeve from the outside end, and proceed from there. But I have not tried it myself. I haven’t yet started wearing long-sleeve shirts for the season.

Bill Amend’s FoxTrot for the 11th — a new strip — does a story problem featuring pizzas cut into some improbable numbers of slices. I don’t say it’s unrealistic someone might get this homework problem. Just that the story writer should really ask whether they’ve ever seen a pizza cut into sevenths. I have a faint memory of being served a pizza cut into tenths by same daft pizza shop, which implies fifths is at least possible. Sevenths I refuse, though.

Mark Tatulli’s Heart of the City for the 12th plays on the show-your-work directive many mathematics assignments carry. I like Heart’s showiness. But the point of showing your work is because nobody cares what (say) 224 divided by 14 is. What’s worth teaching is the ability to recognize what approaches are likely to solve what problems. What’s tested is whether someone can identify a way to solve the problem that’s likely to succeed, and whether that can be carried out successfully. This is why it’s always a good idea, if you are stumped on a problem, to write out how you think this problem should be solved. Writing out what you mean to do can clarify the steps you should take. And it can guide your instructor to whether you’re misunderstanding something fundamental, or whether you just missed something small, or whether you just had a bad day.

Norm Feuti’s Gil for the 12th, another rerun, has another fanciful depiction of showing your work. The teacher’s got a fair complaint in the note. We moved away from tally marks as a way to denote numbers for reasons. Twelve depictions of apples are harder to read than the number 12. And they’re terrible if we need to depict numbers like one-half or one-third. Might be an interesting side lesson in that.

Brian Basset’s Red and Rover for the 14th is a rerun and one I’ve mentioned in these parts before. I understand Red getting fired up to be an animator by the movie. It’s been a while since I watched Donald Duck in Mathmagic Land but my recollection is that while it was breathtaking and visually inventive it didn’t really get at mathematics. I mean, not at noticing interesting little oddities and working out whether they might be true always, or sometimes, or almost never. There is a lot of play in mathematics, especially in the exciting early stages where one looks for a thing to prove. But it’s also in seeing how an ingenious method lets you get just what you wanted to know. I don’t know that the short demonstrates enough of that.

Bud Blake’s Tiger rerun for the 15th gives Punkinhead the chance to ask a question. And it’s a great question. I’m not sure what I’d say arithmetic is, not if I’m going to be careful. Offhand I’d say arithmetic is a set of rules we apply to a set of things we call numbers. The rules are mostly about how we can take two numbers and a rule and replace them with a single number. And these turn out to correspond uncannily well with the sorts of things we do with counting, combining, separating, and doing some other stuff with real-world objects. That it’s so useful is why, I believe, arithmetic and geometry were the first mathematics humans learned. But much of geometry we can see. We can look at objects and see how they fit together. Arithmetic we have to infer from the way the stuff we like to count works. And that’s probably why it’s harder to do when we start school.

What’s not good about that as an answer is that it actually applies to a lot of mathematical constructs, including those crazy exotic ones you sometimes see in science press. You know, the ones where there’s this impossibly complicated tangle with ribbons of every color and a headline about “It’s Revolutionary. It’s 46-Dimensional. It’s Breaking The Rules Of Geometry. Is It The Shape That Finally Quantizes Gravity?” or something like that. Well, describe a thing vaguely and it’ll match a lot of other things. But also when we look to new mathematical structures, we tend to look for things that resemble arithmetic. Group theory, for example, is one of the cornerstones of modern mathematical thought. It’s built around having a set of things on which we can do something that looks like addition. So it shouldn’t be a surprise that many groups have a passing resemblance to arithmetic. Mathematics may produce universal truths. But the ones we see are also ones we are readied to see by our common experience. Arithmetic is part of that common experience.

Also Jerry Scott and Jim Borgman’s Zits for the 14th I think doesn’t really belong here. It’s just got a cameo appearance by the concept of mathematics. Dave Whamond’s Reality Check for the 17th similarly just mentions the subject. But I did want to reassure any readers worried after last week that Pierce recovered fine. Also that, you know, for not having a stomach for mathematics he’s doing well carrying on. Discipline will carry one far.

## Reading the Comics, August 1, 2016: Kalends Edition

The last day of July and first day of August saw enough mathematically-themed comic strips to fill a standard-issue entry. The rest of the week wasn’t so well-stocked. But I’ll cover those comics on Tuesday if all goes well. This may be a silly plan, but it is a plan, and I will stick to that.

Johnny Hart’s Back To BC reprints the venerable and groundbreaking comic strip from its origins. On the 31st of July it reprinted a strip from February 1959 in which Peter discovers mathematics. The work’s elaborate, much more than we would use to solve the problem today. But it’s always like that. Newly-discovered mathematics is much like any new invention or innovation, a rickety set of things that just barely work. With time we learn better how the idea should be developed. And we become comfortable with the cultural assumptions going into the work. So we get more streamlined, faster, easier-to-use mathematics in time.

The early invention of mathematics reappears the 1st of August, in a strip from earlier in February 1959. In this case it’s the sort of word problem confusion strip that any comic with a student could do. That’s a bit disappointing but Hart had much less space than he’d have for the Sunday strip above. One must do what one can.

Mac King and Bill King’s Magic in a Minute for the 31st maybe isn’t really mathematics. I guess there’s something in the modular-arithmetic implied by it. But it depends on a neat coincidence. Follow the directions in the comic about picking a number from one to twelve and counting out the letters in the word for that number. And then the letters in the word for the number you’re pointing to, and then once again. It turns out this leads to the same number. I’d never seen this before and it’s neat that it does.

Rick Detorie’s One Big Happy rerun for the 31st features Ruthie teaching, as she will. She mentions offhand the “friendlier numbers”. By this she undoubtedly means the numbers that are attractive in some way, like being nice to draw. There are “friendly numbers”, though, as number theorists see things. These are sets of numbers. For each number in this set you get the same index if you add together all its divisors (including 1 and the original number) and divide it by the original number. For example, the divisors of six are 1, 2, 3, and 6. Add that together and you get 12; divide that by the original 6 and you get 2. The divisors of 28 are 1, 2, 4, 7, 14, and 28. Add that pile of numbers together and you get 56; divide that by the original 28 and you get 2. So 6 and 28 are friendly numbers, each the friend of the other.

As often happens with number theory there’s a lot of obvious things we don’t know. For example, we know that 1, 2, 3, 4, and 5 have no friends. But we do not know whether 10 has. Nor 14 nor 20. I do not know if it is proved whether there are infinitely many sets of friendly numbers. Nor do I know if it is proved whether there are infinitely many numbers without friends. Those last two sentences are about my ignorance, though, and don’t reflect what number theory people know. I’m open to hearing from people who know better.

There are also things called “amicable numbers”, which are easier to explain and to understand than “friendly numbers”. A pair of numbers are amicable if the sum of one number’s divisors is the other number. 220 and 284 are the smallest pair of amicable numbers. Fermat found that 17,296 and 18,416 were an amicable pair; Descartes found that 9,363,584 and 9,437,056 were. Both pairs were known to Arab mathematicians already. Amicable pairs are easy enough to produce. From the tenth century we’ve had Thâbit ibn Kurrah’s rule, which lets you generate sets of numbers. Ruthie wasn’t thinking of any of this, though, and was more thinking how much fun it is to write a 7.

Terry Border’s Bent Objects for the 1st just missed the anniversary of John Venn’s birthday and all the joke Venn Diagrams that were going around at least if your social media universe looks anything like mine.

Jon Rosenberg’s Scenes from a Multiverse for the 1st is set in “Mathpinion City”, in the “Numerically Flexible Zones”. And I appreciate it’s a joke about the politicization of science. But science and mathematics are human activities. They are culturally dependent. And especially at the dawn of a new field of study there will be long and bitter disputes about what basic terms should mean. It’s absurd for us to think that the question of whether 1 + 1 should equal 2 or 3 could even arise.

But we think that because we have absorbed ideas about what we mean by ‘1’, ‘2’, ‘3’, ‘plus’, and ‘equals’ that settle the question. There was, if I understand my mathematics history right — and I’m not happy with my reading on this — a period in which it was debated whether negative numbers should be considered as less than or greater than the positive numbers. Absurd? Thermodynamics allows for the existence of negative temperatures, and those represent extremely high-energy states, things that are hotter than positive temperatures. A thing may get hotter, from 1 Kelvin to 4 Kelvin to a million Kelvin to infinitely many Kelvin to -1000 Kelvin to -6 Kelvin. If there are intuition-defying things to consider about “negative six” then we should at least be open to the proposition that the universal truths of mathematics are understood by subjective processes.

## Reading the Comics, July 2, 2016: Ripley’s Edition

As I said Sunday, there were more mathematics-mentioning comic strips than I expected last week. So do please read this little one and consider it an extra. The best stuff to talk about is from Ripley’s Believe It Or Not, which may or may not count as a comic strip. Depends how you view these things.

Randy Glasbergen’s Glasbergen Cartoons for the 29th just uses arithmetic as the sort of problem it’s easiest to hide in bed from. We’ve all been there. And the problem doesn’t really enter into the joke at all. It’s just easy to draw.

John Graziano’s Ripley’s Believe It Or Not on the 29th shows off a bit of real trivia: that 599 is the smallest number whose digits add up to 23. And yet it doesn’t say what the largest number is. That’s actually fair enough. There isn’t one. If you had a largest number whose digits add up to 23, you could get a bigger one by multiplying it by ten: 5990, for example. Or otherwise add a zero somewhere in the digits: 5099; or 50,909; or 50,909,000. If we ignore zeroes, though, there are finitely many different ways to write a number with digits that add up to 23. This is almost an example of a partition problem. Partitions are about how to break up a set of things into groups of one or more. But in a partition proper we don’t really care about the order: 5-9-9 is as good as 9-9-5. But we can see some minor differences between 599 and 995 as numbers. I imagine there must be a name for the sort of partition problem in which order matters, but I don’t know what it is. I’ll take nominations if someone’s heard of one.

Graziano’s Ripley’s sneaks back in here the next day, too, with a trivia almost as baffling as the proper credit for the strip. I don’t know what Graziano is getting at with the claim that Ancient Greeks didn’t consider “one” to be a number. None of the commenters have an idea either and my exhaustive minutes of researching haven’t worked it out.

But I wouldn’t blame the Ancient Greeks for finding something strange about 1. We find something strange about it too. Most notably, of all the counting numbers 1 falls outside the classifications of “prime” and “composite”. It fits into its own special category, “unity”. It divides into every whole number evenly; only it and zero do that, if you don’t consider zero to be a whole number. It’s the multiplicative identity, and it’s the numerator in the set of unit fractions — one-half and one-third and one-tenth and all that — the first fractions that people understand. There’s good reasons to find something exceptional about 1.

dro-mo for the 30th somehow missed both Pi Day and Tau Day. I imagine it’s a rerun that the artist wasn’t watching too closely.

Aaron McGruder’s The Boondocks rerun for the 2nd concludes that storyline I mentioned on Sunday about Riley not seeing the point of learning subtraction. It’s always the motivation problem.

## Reading the Comics, June 25, 2016: Busy Week Edition

I had meant to cut the Reading The Comics posts back to a reasonable one a week. Then came the 23rd, which had something like six hundred mathematically-themed comic strips. So I could post another impossibly long article on Sunday or split what I have. And splitting works better for my posting count, so, here we are.

Charles Brubaker’s Ask A Cat for the 19th is a soap-bubbles strip. As ever happens with comic strips, the cat blows bubbles that can’t happen without wireframes and skillful bubble-blowing artistry. It happens that a few days ago I linked to a couple essays showing off some magnificent surfaces that the right wireframe boundary might inspire. The mathematics describing how a soap bubbles’s shape should be made aren’t hard; I’m confident I could’ve understood the equations as an undergraduate. Finding exact solutions … I’m not sure I could have done. (I’d still want someone looking over my work if I did them today.) But numerical solutions, that I’d be confident in doing. And the real thing is available when you’re ready to get your hands … dirty … with soapy water.

Rick Stromoski’s Soup To Nutz for the 19th Shows RoyBoy on the brink of understanding symmetry. To lose at rock-paper-scissors is indeed just as hard as winning is. Suppose we replaced the names of the things thrown with letters. Suppose we replace ‘beats’ and ‘loses to’ with nonsense words. Then we could describe the game: A flobs B. B flobs C. C flobs A. A dostks C. C dostks B. B dostks A. There’s no way to tell, from this, whether A is rock or paper or scissors, or whether ‘flob’ or ‘dostk’ is a win.

Bill Whitehead’s Free Range for the 20th is the old joke about tipping being the hardest kind of mathematics to do. Proof? There’s an enormous blackboard full of symbols and the three guys in lab coats are still having trouble with it. I have long wondered why tips are used as the model of impossibly difficult things to compute that aren’t taxes. I suppose the idea of taking “fifteen percent” (or twenty, or whatever) of something suggests a need for precision. And it’ll be fifteen percent of a number chosen without any interest in making the calculation neat. So it looks like the worst possible kind of arithmetic problem. But the secret, of course, is that you don’t have to have “the” right answer. You just have to land anywhere in an acceptable range. You can work out a fraction — a sixth, a fifth, or so — of a number that’s close to the tab and you’ll be right. So, as ever, it’s important to know how to tell whether you have a correct answer before worrying about calculating it.

Allison Barrows’s Preeteena rerun for the 20th is your cheerleading geometry joke for this week.

I am sure Bill Holbrook’s On The Fastrack for the 22nd is not aimed at me. He hangs around Usenet group rec.arts.comics.strips some, as I do, and we’ve communicated a bit that way. But I can’t imagine he thinks of me much or even at all once he’s done with racs for the day. Anyway, Dethany does point out how a clear identity helps one communicate mathematics well. (Fi is to talk with elementary school girls about mathematics careers.) And bitterness is always a well-received pose. Me, I’m aware that my pop-mathematics brand identity is best described as “I guess he writes a couple things a week, doesn’t he?” and I could probably use some stronger hook, somewhere. I just don’t feel curmudgeonly most of the time.

Darby Conley’s Get Fuzzy rerun for the 22nd is about arithmetic as a way to be obscure. We’ve all been there. I had, at first, read Bucky’s rating as “out of 178 1/3 π” and thought, well, that’s not too bad since one-third of π is pretty close to 1. But then, Conley being a normal person, probably meant “one-hundred seventy-eight and a third”, and π times that is a mess. Well, it’s somewhere around 550 or so. Octave tells me it’s more like 560.251 and so on.

## Reading the Comics, June 11, 2016: Mostly Mathematics As A Signifier Edition

For this week’s roundup of mathematically themed comic strips I have a picture again! After a month or so. It’s great to see again. Also there’s several comics I could swear I’ve shown and featured before. But it’s really quite hot here and I don’t feel like going to the effort of looking. If I repeat myself, so I do. I bet you’ve forgotten the last time I did this Robbie and Bobby too.

Carol Lay’s Lay Lines for the 6th implicitly uses mathematics as an example of perfection. The idea of the straight line is in that territory shared by both mathematics and Platonic ideals. We can imagine a straight line and understand many properties of it even though it can’t be manifest in our real world. The Gods, allegedly, would be able to overcome that and offer perfect circles around imperfect lines. I suppose that’s one way to tell there’s a god involved. The strip also take a moment to riff on the ontological problem, although I don’t know if that’s part of Lay’s intent.

Jonathan Lemon’s Rabbits Against Magic for the 6th uses a bit of mathematics to represent having a theory. It’s true enough that mathematics serves this role in many sciences. We can often put a good explanation for phenomena in a set of equations. But that’s so if you have a good idea what quantities to measure, and how they affect one another. Lettuce’s equation just describes how long an arc within a circle is. It’s true, although I don’t think it rates the status of a theory; it just describes one thing we’d like to know in terms of another thing. And it’s all a setup for a π joke anyway.

Bud Blake’s Tiger for the 8th, as a King Features comic, broke my drought of having images to include with Reading the Comics posts! Celebrate! It’s also the one that made me think I was getting reruns in. But it’s more mysterious than that. The Tiger rerun (Blake died several years ago and all Tiger strips are reruns) for the 20th of April, 2015, is the same joke. I featured it in a Reading The Comics post back then. But it’s not the same strip. The art’s completely redrawn. I can’t fault Blake for having reused a setup-and-punchline. Every comic strip creator does this. Sometimes the cartoonist has improved the joke. (Berkeley Breathed did this several times over.) Sometimes the cartoonist probably just forgot it was done before. (There’s several Peanuts strips suggesting this.) I’m just delighted to catch someone at it.

Ryan Pagelow’s Buni for the 8th uses a blackboard full of mathematics as signifier for explaining the Big Questions of life. And features the traditional little error spotted by someone else. The scribbles are gibberish altogether, but they don’t need to be (and in truth couldn’t be) meaningful. I will defend the backwards-capital-sigma in the upper left of the first panel, though. Sigmas are some of those letters that get pretty sloppy treatment. You get swept up in inspiration and penmanship just collapses. Other Greek letters take some shabby treatment too. And there was a stretch of about three years when I would’ve sworn there was a letter ‘ksee’, a sort of topheavy squiggle. It doesn’t exist, but it’s pretty convenient when you need one more easy Greek letter to use.

Jason Poland’s Robbie and Bobby for the 8th is the second strip that made me think there were reruns. I was right. It ran in September 2014, and I had it then.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 8th features the “scariest equation” in the universe. The board gives a good description of the quantities in the equation and the relationship makes superficial sense. But it does depend on an assumption I’m not sure about, but I will go with. Weinersmith’s argument supposes that a mis-sent text is equally likely to go to any of your contacts. I am not an experienced texter. But it seems to me that a mis-sent text is more likely to go to a contact you’d recently messaged, or one that’s close to the person you meant to contact. Suppose parents are among the people you text often, or whose contact information is stored where it’s easy to pick by accident. Then you likely send them more messages by accident than this expects. On the other hand, suppose you don’t text parents often or you store their information well away from your significant’s. Then the number of mis-sent messages given to them is lower. Without information about how you organize your contacts, we can’t say what’s a better estimate. So in ignorance we may suppose you mis-send texts to every one of your contacts equally often.

Samson’s Dark Side of the Horse for the 9th is the numerals-as-objects joke for this time around.

Dave Whamond’s Reality Check for the 9th uses word problems as the signifier for everything mathematics teachers want to know.

Mell Lazarus’s Momma rerun for the 10th uses a word problem to try to quantify love. Marylou decries the result as “differential calculus”, although it’s really just high school algebra. “Differential calculus” is the funnier term, must admit. Differential calculus refers, generally, to the study of how much one quantity depends on another. On average you can expect something to change if one or more of the variables that describe it change. For example, if you make a rectangle a little larger, its area gets larger. What’s the ratio between how much the area changes and how much the lengths of the rectangle change? If you make an angled corridor wider, then a longer straight object can be fit through the corner. How much longer an object can you bring through the corridor if you make the width a tiny bit bigger? And this also tells us where maximums and minimums are. At a maximum or minimum, a quantity doesn’t change appreciably as the variables that describe it change a little bit. So we can find maximums and minimums by the differential calculus.

Mark Anderson’s Andertoons finally gets in on the 11th. The numbers are looking good. I’m happy with it.

Yeah, so, it wasn’t really all that hot.

## Reading the Comics, May 28, 2016: Visual Interest Will Never Reappear Edition

OK, that’s three weeks in a row in which all my mathematically-themed comic strips are from Gocomics. Maybe I should commission some generic Reading The Comics art from the cartoonists and artists I know. It could make things more exciting on a visually dull week like this.

Mark Anderson’s Andertoons got its entry on the 25th. We draw the name “exponents” from the example of Michael Stifel, a 16th-century German theologian/mathematician. He’d described them as exponents in his influential 1544 book Arithmetica Integra. But I don’t know why he picked the name “exponent” rather than some other word.

Nate Fakes’s Break of Day for the 25th is the anthropomorphic numerals gag for this week.

Dave Blazek’s Loose Parts for the 25th is not quite the anthropomorphic shapes joke for this week. The word “isosceles” does trace back to Greek, of course. The first part comes from “isos”, meaning equal; you see the same root in terms like “isobar” and “isometric view”. The “sceles” part comes from “skelos”, meaning leg. Say what you will about an isosceles triangle, and you may as they’ve got poor hearing, but they do have two legs with the same length. If you want to say an equilateral triangle, which has three legs the same length, is an isosceles triangle you can do that. You’ll be right. But you will look like you’re trying a little too hard to make a point, the way you do if you point to a square and start off by calling it a rhombus.

Donna A Lewis’s Reply All Lite for the 25th tries doing a joke about doing mathematics by hand being a sign of old age. If we’re talking about arithmetic … I could go along with that, grudgingly. Calculator applications are so reliable and so quick that it’s hard to justify doing arithmetic by hand unless it’s a very simple problem. If you have fun doing that, good.

But if we’re doing real mathematics, the working out of a model and the implications of that, or working out calculus or group theory or graph theory or the like? There are surely some people who can do all this work in their heads and I am impressed by that. But much of real mathematics is working out implications of ideas, and that’s done so very well by hand. I haven’t found a way of typing in strings of expressions which makes it easier for me to think about the mathematics rather than the formatting. And I would believe in a note-taking program that was as sensitive and precise as pen on paper. I haven’t seen one yet, though. (I have small handwriting, and the applications I’ve tried turn all my writing into tiny, disconnected dots and scribbles.)

Ralph Hagen’s The Barn for the 27th is superficially about Olbers’s Paradox. If there’s an infinitely large, infinitely old universe, then how can the night sky by dark? The light of all those stars should come together to make night even more brilliantly blazing than the daytime sun. This is a legitimate calculus problem. The reasoning is sound. The light of a star trillions upon trillions of light-years away may be impossibly faint. But there are so many stars that would be that far away that they would be, on average, about as bright as the sun is. Integral calculus tells us what happens when we have infinitely large numbers of impossibly tiny things added together. In the case of stars, infinitely many impossibly faint stars would come together to an infinitely bright night sky. That night is dark tells us: the universe can’t be infinitely large and infinitely old. There must be limits to how far away anything can be.

The Barn reappears in my attention on the 28th, with a subverted word problem joke.

## Reading the Comics, May 17, 2016: Again, No Pictures Edition

Last week’s Reading The Comics was a bunch of Gocomics.com strips. And I don’t feel the need to post the images for those, since they’re reasonably stable links. Today’s is also a bunch of Gocomics.com strips. I know how every how-to-bring-in-readers post ever says you should include images. Maybe I will commission someone to do some icons. It couldn’t hurt.

Someone looking close at the title, with responsible eye protection, might notice it’s dated the 17th, a day this is not. There haven’t been many mathematically-themed comic strips since the 17th is all. And I’m thinking to try out, at least for a while, making the day on which a Reading the Comics post is issued regular. Maybe Monday. This might mean there are some long and some short posts, but being a bit more scheduled might help my writing.

Mark Anderson’s Andertoons for the 14th is the charting joke for this essay. Also the Mark Anderson joke for this essay. I was all ready to start explaining ways that the entropy of something can decrease. The easiest way is by expending energy, which we can see as just increasing entropy somewhere else in the universe. The one requiring the most patience is simply waiting: entropy almost always increases, or at least doesn’t decrease. But “almost always” isn’t the same as “always”. But I have to pass. I suspect Anderson drew the chart going down because of the sense of entropy being a winding-down of useful stuff. Or because of down having connotations of failure, and the increase of entropy suggesting the failing of the universe. And we can also read this as a further joke: things are falling apart so badly that even entropy isn’t working like it ought. Anderson might not have meant for a joke that sophisticated, but if he wants to say he did I won’t argue it.

Scott Adams’s Dilbert Classics for the 14th reprinted the comic of the 20th of March, 1993. I admit I do this sort of compulsive “change-simplifying” paying myself. It’s easy to do if you have committed to memory pairs of numbers separated by five: 0 and 5, 1 and 6, 2 and 7, and so on. So if I get a bill for (say) $4.18, I would look for whether I have three cents in change. If I have, have I got 23 cents? That would give me back a nickel. 43 cents would give me back a quarter in change. And a quarter is great because I can use that for pinball. Sometimes the person at the cash register doesn’t want a ridiculous bunch of change. I don’t blame them. It’s easy to suppose that someone who’s given you$5.03 for a $4.18 charge misunderstood what the bill was. Some folks will take this as a chance to complain mightily about how kids don’t learn even the basics of mathematics anymore and the world is doomed because the young will follow their job training and let machines that are vastly better at arithmetic than they are do arithmetic. This is probably what Adams was thinking, since, well, look at the clerk’s thought balloon in the final panel. But consider this: why would Dilbert have handed over$7.14? Or, specifically, how could he give $7.14 to the clerk but not have been able to give$2.14, which would make things easier on everybody? There’s no combination of bills — in United States or, so far as I’m aware, any major world currency — in which you can give seven dollars but not two dollars. He had to be handing over five dollars he was getting right back. The clerk would be right to suspect this. It looks like the start of a change scam, begun by giving a confusing amount of money.

Had Adams written it so that the charge was $6.89, and Dilbert “helpfully” gave$12.14, then Dilbert wouldn’t be needlessly confusing things.

Dave Whamond’s Reality Check for the 15th is that pirate-based find-x joke that feels like it should be going around Facebook, even though I don’t think it has been. I can’t say the combination of jokes quite makes logical sense, but I’m amused. It might be from the Reality Check squirrel in the corner.

Nate Fakes’s Break of Day for the 16th is the anthropomorphized shapes joke for this essay. It’s not the only shapes joke, though.

Doug Bratton’s Pop Culture Shock Therapy for the 16th is the Einstein joke for this essay.

Rick Detorie’s One Big Happy rerun for the 17th is another shapes joke. Ruthie has strong ideas about what distinguishes a pyramid from a triangle. In this context I can’t say she’s wrong to assert what a pyramid is.

## Reading the Comics, May 12, 2016: No Pictures Again Edition

I’ve hardly stopped reading the comics. I doubt I could even if I wanted at this point. But all the comics this bunch are from GoComics, which as far as I’m aware doesn’t turn off access to comic strips after a couple of weeks. So I don’t quite feel justified including the images of the comics when you can just click links to them instead.

It feels a bit barren, I admit. I wonder if I shouldn’t commission some pictures so I have something for visual appeal. There’s people I know who do comics online. They might be able to think of something to go alongside every “Student has snarky answer for a word problem” strip.

Brian and Ron Boychuk’s The Chuckle Brothers for the 8th of May drops in an absolute zero joke. Absolute zero’s a neat concept. People became aware of it partly by simple extrapolation. Given that the volume of a gas drops as the temperature drops, is there a temperature at which the volume drops to zero? (It’s complicated. But that’s the thread I use to justify pointing out this strip here.) And people also expected there should be an absolute temperature scale because it seemed like we should be able to describe temperature without tying it to a particular method of measuring it. That is, it would be a temperature “absolute” in that it’s not explicitly tied to what’s convenient for Western Europeans in the 19th century to measure. That zero and that instrument-independent temperature idea get conflated, and reasonably so. Hasok Chang’s Inventing Temperature: Measurement and Scientific Progress is well-worth the read for people who want to understand absolute temperature better.

Gene Weingarten, Dan Weingarten & David Clark’s Barney and Clyde for the 9th is another strip that seems like it might not belong here. While it’s true that accidents sometimes lead to great scientific discoveries, what has that to do with mathematics? And the first thread is that there are mathematical accidents and empirical discoveries. Many of them are computer-assisted. There is something that feels experimental about doing a simulation. Modern chaos theory, the study of deterministic yet unpredictable systems, has at its founding myth Edward Lorentz discovering that tiny changes in a crude weather simulation program mattered almost right away. (By founding myth I don’t mean that it didn’t happen. I just mean it’s become the stuff of mathematics legend.)

But there are other ways that “accidents” can be useful. Monte Carlo methods are often used to find extreme — maximum or minimum — solutions to complicated systems. These are good if it’s hard to find a best possible answer, but it’s easy to compare whether one solution is better or worse than another. We can get close to the best possible answer by picking an answer at random, and fiddling with it at random. If we improve things, good: keep the change. You can see why this should get us pretty close to a best-possible-answer soon enough. And if we make things worse then … usually but not always do we reject the change. Sometimes we take this “accident”. And that’s because if we only take improvements we might get caught at a local extreme. An even better extreme might be available but only by going down an initially unpromising direction. So it’s worth allowing for some “mistakes”.

Mark Anderson’s Andertoons for the 10th of Anderson is some wordplay on volume. The volume of boxes is an easy formula to remember and maybe it’s a boring one. It’s enough, though. You can work out the volume of any shape using just the volume of boxes. But you do need integral calculus to tell how to do it. So maybe it’s easier to memorize the formula for volumes of a pyramid and a sphere.

Berkeley Breathed’s Bloom County for the 10th of May is a rerun from 1981. And it uses a legitimate bit of mathematics for Milo to insult Freida. He calls her a “log 10 times 10 to the derivative of 10,000”. The “log 10” is going to be 1. A reference to logarithm, without a base attached, means either base ten or base e. “log” by itself used to invariably mean base ten, back when logarithms were needed to do ordinary multiplication and division and exponentiation. Now that we have calculators for this mathematicians have started reclaiming “log” to mean the natural logarithm, base e, which is normally written “ln”, but that’s still an eccentric use. Anyway, the logarithm base ten of ten is 1: 10 is equal to 10 to the first power.

10 to the derivative of 10,000 … well, that’s 10 raised to whatever number “the derivative of 10,000” is. Derivatives take us into calculus. They describe how much a quantity changes as one or more variables change. 10,000 is just a number; it doesn’t change. It’s called a “constant”, in another bit of mathematics lingo that reminds us not all mathematics lingo is hard to understand. Since it doesn’t change, its derivative is zero. As anything else changes, the constant 10,000 does not. So the derivative of 10,000 is zero. 10 to the zeroth power is 1.

So, one times one is … one. And it’s rather neat that kids Milo’s age understand derivatives well enough to calculate that.

Ruben Bolling’s Super-Fun-Pak Comix rerun for the 10th happens to have a bit of graph theory in it. One of Uncle Cap’n’s Puzzle Pontoons is a challenge to trace out a figure without retracting a line or lifting your pencil. You can’t, not this figure. One of the first things you learn in graph theory teaches how to tell, and why. And thanks to a Twitter request I’m figuring to describe some of that for the upcoming Theorem Thursdays project. Watch this space!

Charles Schulz’s Peanuts Begins for the 11th, a rerun from the 6th of February, 1952, is cute enough. It’s one of those jokes about how a problem seems intractable until you’ve found the right way to describe it. I can’t fault Charlie Brown’s thinking here. Figuring out a way the problems are familiar and easy is great.

Shaenon K Garrity and Jeffrey C Wells’s Skin Horse for the 12th is a “see, we use mathematics in the real world” joke. In this case it’s triangles and triangulation. That’s probably the part of geometry it’s easiest to demonstrate a real-world use for, and that makes me realize I don’t remember mathematics class making use of that. I remember it coming up some, particularly in what must have been science class when we built and launched model rockets. We used a measure of how high an angle the rocket reached, and knowledge of how far the observing station was from the launchpad. But that wasn’t mathematics class for some reason, which is peculiar.

## Reading the Comics, May 6, 2016: Mistakes Edition

I knew my readership would drop off after I fell back from daily posting. Apparently it was worse than I imagined and nobody read my little blog here over the weekend. That’s fair enough; I had to tend other things myself. Still, for the purpose of maximizing the number of page views around here, taking two whole days off in a row was a mistake. There’s some more discussed in this Reading The Comics installment.

Word problems are dull. At least at the primary-school level. There’s all these questions about trains going in different directions or ropes sweeping out areas or water filling troughs. So Aaron McGruder’s Boondocks rerun from the 5th of May (originally run the 22nd of February, 2001) is a cute change. It’s at least the start of a legitimate word problem, based on the ways the recording industry took advantage of artists in the dismal days of fifteen years ago. I’m sure that’s all been fixed by now. Fill in some numbers and the question might interest people.

Glenn McCoy and Gary McCoy’s The Duplex for the 5th of May is a misunderstanding-fractions joke. I’m amused by the idea of messing up quarter-pound burgers. But it also brings to mind a summer when I worked for the Great Adventure amusement park and got assigned one day as cashier at the Great American Hamburger Stand. Thing is, I didn’t know anything about the stand besides the data point that they probably sold hamburgers. So customers would order stuff I didn’t know, and I couldn’t find how to enter it on the register, and all told it was a horrible mess. If you were stuck in that impossibly slow-moving line, I am sorry, but it was management’s fault; I told them I didn’t know what I was even selling. Also I didn’t know the drink cup sizes so I just charged you for whatever you said and if I gave you the wrong size I hope it was more soda than you needed.

On a less personal note, I have heard the claim about why one-third-pound burgers failed in United States fast-food places. Several chains tried them out in the past decade and they didn’t last, allegedly because too many customers thought a third of a pound was less than a quarter pound and weren’t going to pay more for less beef. It’s … plausible enough, I suppose, because people have never been good with fractions. But I suspect the problem is more linguistic. A quarter-pounder has a nice rhythm to it. A half-pound burger is a nice strong order to say. A third-pound burger? The words don’t even sound right. You have to say “third-of-a-pound burger” to make it seem like English, and it’s a terribly weak phrase. The fast food places should’ve put their money into naming it something that suggested big-ness but not too-big-to-eat.

Mark Tatulli’s Heart of the City for the 5th is about Heart’s dread of mathematics. Her expressed fear, that making one little mistake means the entire answer is wrong, is true enough. But how how much is that “enough”? If you add together someting that should be (say) 18, and you make it out to be 20 instead, that is an error. But that’s a different sort of error from adding them together and getting 56 instead.

And errors propagate. At least they do in real problems, in which you are calculating something because you want to use it for something else. An arithmetic error on one step might grow, possibly quite large, with further steps. That’s trouble. This is known as an “unstable” numerical calculation, in much the way a tin of picric acid dropped from a great height onto a fire is an “unstable” chemical. The error might stay about as large as it started out being, though. And that’s less troublesome. A mistake might stay predictable. The calculation is “stable” In a few blessed cases an error might be minimized by further calculations. You have to arrange the calculations cleverly to make that possible, though. That’s an extremely stable calculation.

And this is important because we always make errors. At least in any real calculation we do. When we want to turn, say, a formula like πr2 into a number we have to make a mistake. π is not 3.14, nor is it 3.141592, nor is it 3.14159265358979311599796346854418516. Does the error we make by turning π into some numerical approximation matter? It depends what we’re calculating, and how. There’s no escaping error and it might be a comfort to Heart, or any student, to know that much of mathematics is about understanding and managing error.

Joe Martin’s Boffo for the 6th of May is in its way about the wonder of very large numbers. On some reasonable assumptions — that our experience is typical, that nothing is causing traits to be concentrated one way or another — we can realize that we probably will not see any extreme condition. In this case, it’s about the most handsome men in the universe probably not even being in our galaxy. If the universe is large enough and people common enough in it, that’s probably right. But we likely haven’t got the least handsome either. Lacking reason to suppose otherwise we can guess that we’re in the vast middle.

David L Hoyt and Jeff Knurek’s Jumble for the 6th of May mentions mathematicians and that’s enough, isn’t it? Without spoiling the puzzle for anyone, I will say that “inocci” certainly ought to be a word meaning something. So get on that, word-makers.

Dave Blazek’s Loose Parts for the 6th brings some good Venn Diagram humor back to my pages. Good. It’s been too long.

## Reading the Comics, May 3, 2016: Lots Of Images Edition

After the heavy pace of March and April I figure to take it easy and settle to about a three-a-week schedule around here. That doesn’t mean that Comic Strip Master Command wants things to be too slow for me. And this time they gave me more comics than usual that have expiring URLs. I don’t think I’ve had this many pictures to include in a long while.

Bill Whitehead’s Free Range for the 28th presents an equation-solving nightmare. From my experience, this would be … a great pain, yes. But it wouldn’t be a career-wrecking mess. Typically a problem that’s hard to solve is hard because you have no idea what to do. Given an expression, you’re allowed to do anything that doesn’t change its truth value. And many approaches might look promising without quite resolving to something useful. The real breakthrough is working out what approach should be used. For an astrophysics problem, there are some classes of key decisions to make. One class is what to include and what to omit in the model. Another class is what to approximate — and how — versus what to treat exactly. Another class is what sorts of substitutions and transformations turn the original expression into one that reveals what you want. Those are the hard parts, and those are unlikely to have been forgotten. Applying those may be tedious, and I don’t doubt it would be anguishing to have the finished work wiped out. But it wouldn’t set one back years either. It would just hurt.

Christopher Grady’s Lunar Babboon for the 29th I classify as the “anthropomorphic numerals” joke for this essay. Boy, have we all been there.

Bill Holbrook’s On The Fastrack for the 29th continues the storyline about Fi giving her STEM talk. She is right, as I see it, in attributing drama and narrative to numbers. This is most easily seen in the sorts of finance and accounting mathematics which the character does. And the inevitable answer to “numbers are boring” (or “mathematics is boring”) is surely to show how they are about people. Even abstract mathematics is about things (some) people find interesting, and that must be about the people too.

Rick Detorie’s One Big Happy for the 16th is a confused-mathematics joke. Grandpa tosses off a New Math joke that’s reasonably age-appropriate too, which is always nice to see in a comic strip. I don’t know how seriously to take Ruthie’s assertion that a 100% means she only got at least half of the questions correct. It could be a cartoonist grumbling about how kids these days never learn anything, the same way ever past generation of cartoonists had complained. But Ruthie is also the sort of perpetually-confused, perpetually-confusing character who would get the implications of a 100% on a test wrong. Or would state them weirdly, since yes, a 100% does imply getting at least half the test’s questions right.

Niklas Eriksson’s Carpe Diem for the 3rd uses the traditional board full of mathematical symbols as signifier of intelligence. There’s some interesting mixes of symbols here. The c2, for example, isn’t wrong for mathematics. But it does evoke Einstein and physics. There’s the curious mix of the symbol π and the approximation 3.14. But then I’m not sure how we would get from any of this to a proposition like “whether we can survive without people”.

Bud Blake’s Tiger for the 3rd is a cute little kids-learning-to-count thing. I suppose it doesn’t really need to be here. But Punkinhead looks so cute wearing his tie dangling down onto the floor, the way kids wear their ties these days.

Tony Murphy’s It’s All About You for the 3rd name-drops algebra. I think what the author really wanted here was arithmetic, if the goal is to figure out the right time based on four clocks. They seem to be trying to do a simple arithmetic mean of the time on the four clocks, which is fair if we make some assumptions about how clocks drift away from the correct time. Mostly those assumptions are that the clocks all started right and are equally likely to drift backwards or forwards, and do that drifting at the same rate. If some clocks are more reliable than others, then, their claimed time should get more weight than the others. And something like that must be at work here. The mean of 7:56, 8:02, 8:07, and 8:13, uncorrected, is 8:04 and thirty seconds. That’s not close enough to 8:03 “and five-eighths” unless someone’s been calculating wrong, or supposing that 8:02 is more probably right than 8:13 is.