## 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, 2017. I have to agree the border collies haven’t worked out the point of fetch. I also question whether they’ve worked out the simple ballistics of the tossed stick. If the variables mean what they suggest they mean, then dimensional analysis suggests they’ve got at least three fiascos going on here. Maybe they have an idiosyncratic use for variables like ‘v’.

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.

Mort Walker and Dik Browne’s Hi and Lois for the 10th of February, 1960 and rerun the 8th of August, 2017. Remember: you’re only young once, but you can be geopolitically naive forever!

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.

• #### ivasallay 5:34 pm on Monday, 31 July, 2017 Permalink | Reply

I think you covered arithmetic in a very clear, scholarly way.

When I was in the early elementary grades, we didn’t study math. We studied arithmetic.

Here’s a couple more things some people might not know about arithmetic:
1) How to remember the proper spelling of arithmetic: A Rat In The House May Eat The Ice Cream.
2) How to pronounce arithmetic: https://www.quora.com/Why-does-the-pronunciation-of-arithmetic-depend-on-context

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• #### Joseph Nebus 6:27 pm on Wednesday, 2 August, 2017 Permalink | Reply

Thanks! … My recollection is that in elementary school we called it mathematics (or just math), but the teachers were pretty clear about whether we were doing arithmetic or geometry. If that was clear, since I grew up on the tail end of the New Math wave and we could do stuff that was more playful than multiplication tables were.

I hadn’t thought about the shifting pronunciations of ‘arithmetic’ as a word. I suppose it’s not different from many multi-syllable words in doing that, though. My suspicion is that the distinction between ‘arithmetic’ as an adjective and as a noun is spurious, though. My hunch is people shift the emphasis based on the structure of the whole sentence, with the words coming after ‘arithmetic’ having a big role to play. I’d expect that an important word immediately follows ‘arithmetic’ often if it’s being used as an adjective (like, ‘arithmetic mean’), but that’s not infallible. As opposed to those many rules of English grammar and pronunciation that are infallible.

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• #### gaurish 9:48 am on Saturday, 12 August, 2017 Permalink | Reply

A Beautiful introduction to Arithmetic!

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• #### Joseph Nebus 5:06 pm on Sunday, 13 August, 2017 Permalink | Reply

Thank you kindly. I had a great time writing it.

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## 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 of July, 2017. First, if anyone isn’t thinking of that Pixar short then I’m not sure we can really understand each other. Second, ‘von Thump’ is a fine name for a bunny scientist and if it wasn’t ever used in the rich lore of Usenet group alt.devilbunnies I shall be disappointed. Third, Eriksson made an understandable but unfortunate mistake in composing this panel. While both rabbits are wearing glasses, they’re facing away from the viewer. It’s always correct to draw animals wearing eyeglasses, or to photograph them so. But we should get to see them in full eyeglass pelage. You’d think they would teach that in Cartoonist School or something.

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?

Vic Lee’s Pardon My Planet for the 17th of July, 2017. I am surprised she had no questions about how many dimes Jonathan must have, although perhaps that will follow obviously from knowing the Beth nickel situation.

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.

• #### goldenoj 8:05 pm on Sunday, 23 July, 2017 Permalink | Reply

Human Cull may be the most disturbing idea for a comic ever.

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• #### Joseph Nebus 3:37 am on Tuesday, 25 July, 2017 Permalink | Reply

It’s a strip that has really, deeply bothered me. I know the cartoonist is talking about “culling” in a comically great overreaction to people who might be a bit annoying in your daily life. But that started out feeling terribly nasty when we’re talking about, like, people who put their jackets over empty seats next to them at the movie theater. And given the turn the world’s taken for the nasty the past couple years it hasn’t improved the strip’s tone any.

It’s not like “here’s an annoying thing people do” is an inherently bad idea for a comic strip. It drove They’ll Do It Every Time for much of its run, and it underlay comic strips like The Dinette Set or (in its implications) Pluggers. But I think there needs to be a bit more clearly expressed empathy and grace for the approach to really work.

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• #### The Chaos Realm 3:42 pm on Monday, 24 July, 2017 Permalink | Reply

Now they have a machine that counts out the tills at shift changes or for money drops–I just saw it in action at a local grocery store. The “good old days” (aka massive headache) of counting it out by hand or trying to make the tills balance out are over, it seems. Whew! LOL

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• #### Joseph Nebus 3:47 am on Tuesday, 25 July, 2017 Permalink | Reply

Those have turned up, where I buy stuff, sporadically over the last couple decades. I haven’t worked out what the rhyme or reason for a particular shop having one is, though. It’s not all the shops in a chain and it doesn’t seem to be particularly tied to how urban or rural the place is or how new the shop is. Maybe it’s tied to how often the management thinks cashiers are pocketing change and that’s too idiosyncratic for a mere customer to know.

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laugh

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## 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 of June, 2017. So, one of my retail stories that I might well have already told because I only ever really had one retail job and there’s only so many stories you get working a year and a half in a dying mall’s book store. I was a clerk at Walden Books. The customer wanted to know for this book whether the sticker’s 10 percent discount was taken before or after the state’s 6 percent sales tax was applied. I said I thought the discount taken first and then tax applied, but it didn’t matter if I were wrong as the total would be the same amount. I calculated what it would be. The customer was none too sure about this, but allowed me to ring it up. The price encoded in the UPC was wrong, something like a dollar more than the cover price, and the subtotal came out way higher. The customer declared, “See?” And wouldn’t have any of my explaining that he was hit by a freak event. I don’t remember other disagreements between the UPC price and the cover price, but that might be because we just corrected the price and didn’t get a story out of it.

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. • #### Joseph Nebus 6:00 pm on Wednesday, 31 May, 2017 Permalink | Reply Tags: arithmetic, calculus ( 84 ), infinite series ( 6 ), John Wallis ( 2 ), series ( 4 ) ## 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. • #### elkement (Elke Stangl) 4:38 pm on Monday, 5 June, 2017 Permalink | Reply It reminds me of the famous story about young Gauss – when he baffled his teacher with this somewhat related ‘trick’ of adding up numbers between 1 to 100 very quickly (by actually calculating 101*50). Like • #### Joseph Nebus 1:09 am on Wednesday, 7 June, 2017 Permalink | Reply That’s exactly what crossed my mind, especially as I realized I was doing the sum of 1 through 100 at least implicitly. It feels so playful to have something like that turn up. Liked by 1 person • #### Joseph Nebus 6:00 pm on Sunday, 14 May, 2017 Permalink | Reply Tags: Agnes ( 3 ), arithmetic, Ben, Chuckle Brothers ( 4 ), fractions ( 20 ), Real Life Adventures ( 6 ) ## 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. • #### Joshua K. 4:53 am on Thursday, 18 May, 2017 Permalink | Reply Perhaps the father in the “Ben” strip, rather than snoring, was telling his son about the set of integers. Like • #### Joseph Nebus 6:00 pm on Tuesday, 2 May, 2017 Permalink | Reply Tags: 9 to 5, arithmetic, Between Friends, FoxTrot ( 10 ), Moderately Confused ( 2 ), story problems ( 4 ), Venn diagrams ( 17 ) ## 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 of April, 2017. And while it’s not a Venn Diagram I’m not sure of a better way to visually represent that the cartoonist is going for. I suppose the intended meaning comes across cleanly enough and that’s the most important thing. It’s a strange state of affairs is all. 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. • #### Joseph Nebus 6:00 pm on Thursday, 6 April, 2017 Permalink | Reply Tags: arithmetic, Bloom County ( 3 ), Imogen Quest ( 3 ), intelligence ( 3 ), Mandelbrot Sets ( 2 ), Sticky Comics, Tough Town ( 2 ), Wizard of Id ( 2 ) ## 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. • #### elkement (Elke Stangl) 3:24 pm on Thursday, 20 April, 2017 Permalink | Reply I am replying to the previous post (March statistics) – as nothing happened when I clicked on the reply button at that post. But maybe this is related to what I actually wanted to comment about: Your table is displayed at the bottom of the page – below ‘Related’, the comment box, and the previous/next posting links! How did you do this? You totally hacked WordPress ;-) Like • #### elkement (Elke Stangl) 3:25 pm on Thursday, 20 April, 2017 Permalink | Reply OK – so that reply could be posted. As I said, with your table you confused WordPress a lot :-) Like • #### Joseph Nebus 2:44 am on Tuesday, 25 April, 2017 Permalink | Reply I’m just surprised it’s so easy to confuse it! Liked by 1 person • #### Joseph Nebus 2:43 am on Tuesday, 25 April, 2017 Permalink | Reply Huh, and that’s curious. I didn’t realize it and must not have looked close enough at the preview. It looks like the fault is that I failed to close the table tag, so WordPress tried to fit the rest of the page in-between the tbody and the end of the table and goodness knows how it worked out that presentation. Liked by 1 person • #### Joseph Nebus 6:00 pm on Sunday, 2 April, 2017 Permalink | Reply Tags: Andertoons ( 12 ), arithmetic, Baldo, Big Top ( 2 ), Retail ( 5 ), Tough Town ( 2 ) ## 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 of March, 2017. Of course customers aren’t generally good at arithmetic either. I’m reminded (once more) of when I worked at Walden Books and a customer wanted to know whether the sticker-promised 10 percent discount on the book was applied to the price before or after the 6 percent sales tax was added to it, or whether it was applied afterwards. I could not speak to the cash register’s programming, but I could promise that the process would come to the same number either way, and I told him what it would be. I think the book had a$14.95 cover price — let’s stipulate it was for the sake of my anecdote — so it would come to $14.26 in the end. He judged me suspiciously and then allowed me to ring it up; the register made it out to be$15.22 and he pounced, saying, see?. Yes: he had somehow found the one freaking book in the store where the UPC bar code price, $15.95, was different from the thing listed as the cover price. I told him why it was and showed him where in the UPC to find the encoded price (it’s in the last stanza of digits underneath the bars) but he was having none of it, even when I manually corrected the error. 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. • #### The Chaos Realm 1:05 pm on Monday, 3 April, 2017 Permalink | Reply Forget school-taught math, that’s how I best learned math…as a cashier… Like • #### Joseph Nebus 2:18 am on Tuesday, 4 April, 2017 Permalink | Reply I shouldn’t be surprised! Doing anything often will encourage people to find more accurate and faster ways to do it. So one speeds up either by just being better at recognizing common operations or by developing useful shortcuts. (The shortcuts can be disastrous if, for example, they accidentally cause some needed safety precaution not to be taken, but that doesn’t tend to apply in cashier work.) Liked by 1 person • #### The Chaos Realm 2:29 am on Tuesday, 4 April, 2017 Permalink | Reply Yeah, I used to drive my math teachers crazy with my shortcuts. But, I love when I see the light bulb go off in kids when I show them other ways to do math problems (even as a sub, I do sometimes get to teach :-) ) . Like • #### Joseph Nebus 5:23 am on Friday, 14 April, 2017 Permalink | Reply There is that. A weird shortcut or novel trick for a problem, even if it doesn’t lead to a generally useful technique, is good to have on the record. It inspires the imagination and lets folks know that there’s almost never just one way to do things. Liked by 1 person • #### davekingsbury 9:10 pm on Monday, 3 April, 2017 Permalink | Reply Guestimation keeps the common sense in maths I, er … guess. As for Sudoku, is there any other way to do it than listing all possible #s in each box? I see people on buses and trains just staring at it – are they hoping for inspiration or else doing prodigious memory work? Like • #### Joseph Nebus 2:23 am on Tuesday, 4 April, 2017 Permalink | Reply I’m not an expert sudoku solver. I’d done some for a little while, especially after some students gave me a book of puzzles as a parting gift, but I never caught the bug. But when I do them, it is … I wouldn’t say a prodigious amount of memory work. It would be picking out a cell and checking what the valid possible numbers are, then going across the row, column, and cell to see if there were any obvious contradictions, or whether that forced something suspicious in a nearby cell. I don’t suppose that works well for hard puzzles, but for the silly little easy and almost-medium puzzles I attacked it was fine. Something would turn up soon. Liked by 1 person • #### Joseph Nebus 6:00 pm on Sunday, 26 March, 2017 Permalink | Reply Tags: arithmetic, Dark Side Of The Horse ( 8 ), fractions ( 20 ), Moderately Confused ( 2 ), Pardon My Planet ( 3 ), Pearls Before Swine ( 2 ), Pythagoras ( 9 ), Ripley's Believe It Or Not ( 5 ), sudoku ( 12 ), Todd the Dinosaur ## 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. Patrick Roberts’s Todd the Dinosaur for the 20th of March, 2017. I’ll allow the kids-say-the-darndest-things setup for the strip. I’m stuck on wondering just how much good water wings that size could do. Yes, he’s limited by his anatomy but aren’t we all? 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. Vic Lee’s Pardon My Planet for the 23rd of March, 2017. Don’t mind me, I’m busy trying to convince myself the back left leg of that park bench is hidden behind the guy’s leg and not missing altogether and it’s still pretty touch-and-go on that. 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. • #### Joseph Nebus 6:00 pm on Thursday, 16 March, 2017 Permalink | Reply Tags: Andertoons ( 12 ), arithmetic, Dinosaur Comics ( 2 ), Dustin, On The Fastrack ( 4 ), Red and Rover, SAT ( 3 ), weddings ## 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, 2017. It’s the title character doing the guessing there. Also, Kelley and Parker hate their title character with a thoroughness you rarely see outside Tom Batiuk and Funky Winkerbean. This is a mild case of it but, there we are. 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 of March, 2017. I confess I’m surprised Holbrook didn’t think to set the climax a couple of days later and tie it in to Pi Day. 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. • #### Joseph Nebus 6:00 pm on Thursday, 9 February, 2017 Permalink | Reply Tags: arithmetic, counting ( 11 ), Dark Side Of The Horse ( 8 ), Dogs of C Kennel, entropy ( 29 ), FoxTrot ( 10 ), lottery ( 8 ), Nest Heads, One Big Happy ( 2 ) ## 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 of February, 2017. A ‘gazillion’ is actually a surprisingly low number, hovering as it does somewhere around 212. Fun fact!

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, 2017. You know even for a fundraising event \$17.50 seems a bit much for a hot dog and bottled water. Maybe the friend’s 8-year-old child is way off too.

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.

Gordon Bess’s Redeye for the 21st of September, 1970. Rerun the 1st of February, 2017. I don’t see why they’re so worried about counting bullets if being shot just leaves you a little discombobulated.

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 of January, 2017. Not sure why Shady Shrew is selling the circular wands at 50 cents. Sure, I understand wanting a triangle or star or other wand selling at a premium. But then why have the circular wands at such a cheap price? Wouldn’t it be better to put them at like six dollars, so that eight dollars for a fancy wand doesn’t seem that great an extravagance? You have to consider setting an appropriate anchor point for your customer base. But, then, Shady Shrew isn’t supposed to be that smart.

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.

Gordon Bess’s Redeye rerun from the 28th of August, 1970. Reprinted the 9th of January, 2017. What makes the strip work is how it’s tied to the personalities of these kids and couldn’t be transplanted into every other comic strip with two kids in it.

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 rerun from the 29th of August, 1970. Reprinted the 10th of January, 2017. To be less snarky, I do like the simply-expressed weariness on the girl’s face. It’s hard to communicate feelings with few pen strokes.

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.

• #### gaurish 9:10 am on Saturday, 10 December, 2016 Permalink | Reply

Thanks! When I first learnt about quotient groups (two years ago) I visualized them as the equivalence classes we create so as to have a better understanding of a bigger group (since my study of algebra has been motivated by its need in Number theory as a generalization of modulo arithmetic). Then the isomorphism theorems just changed the way I look at quotient of an algebraic structure. See: http://math.stackexchange.com/q/1816921/214604

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• #### Joseph Nebus 5:47 am on Saturday, 17 December, 2016 Permalink | Reply

I’m glad that you liked. I do think equivalence classes are the easiest way into quotient groups — it’s essentially what I did here — but that’s because people get introduced to equivalence classes without knowing what they are. Odd and even numbers, for example, or checking arithmetic by casting out nines are making use of these classes. Isomorphism theorems are great and substantial but they do take so much preparation to get used to. Probably shifting from the first to the second is the sign of really mastering the idea of a quotient group.

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## 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 of November, 2016. I’m surprised Bell-Lundy used the broader space of a Sunday strip for a joke that doesn’t need that much illustration, but I understand sometimes you just have to go with the joke that you have. And it isn’t as though Sunday comics get that much space anymore either. Anyway, I suppose we have all been there, although for me that’s more often because I used to have a six-digit pin, and a six-digit library card pin, and those were just close enough to each other that I could never convince myself I was remembering the right one in context, so I would guess wrong.

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.

Patrick Roberts’s Todd the Dinosaur for the 7th of November, 2016. I don’t remember being talked to by classmates’ parents about what they where, but that might just be that it’s been a long time since I was in elementary school and everybody had the normal sorts of jobs that kids don’t understand. I guess we talked about what our parents did but that should make a weaker impression.

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.

• #### davekingsbury 10:38 pm on Monday, 14 November, 2016 Permalink | Reply

I reckon it was Bob Newhart’s sketch about it that made the monkey idea so popular. Best bit, something like, hey one of them has something over here er to be or not to be that is the … gezoinebplatf!

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• #### Joseph Nebus 3:35 am on Sunday, 20 November, 2016 Permalink | Reply

I like to think that helped. I fear that that particular routine’s been forgotten, though. I was surprised back in the 90s when I was getting his albums and ran across that bit, as I’d never heard it before. But it might’ve been important in feeding the idea to other funny people. There’s probably a good essay to be written tracing the monkeys at typewriters through pop culture.

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## 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.

David L Hoyt and Jeff Knurek’s Jumble for the 19th of October, 2016. The link will probably expire in about a month. Have to say, it’s not a big class. I’m not surprised the students are doing well.

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.

Rick Kirkman and Jerry Scott’s Baby Blues for the 22nd of October, 2016. I’m not quite curious enough to look, but do wonder how far into the comments you have to go before someone slags on the Common Core. But then I would say if Hammy were to write down first an initial-impression guess of about what the answer should be — say, that “37 + 42” should be a number somewhere around 80 — and then an exact answer, then that would be consistent with what I understand Common Core techniques encourage and a pretty solid approach.

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 of September, 2016. I think Randolph Itch, 2am did this joke too but then had everyone retire to the bar chart.

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.

David L Hoyt and Jeff Knurek’s Jumble for the 1st of October, 2016. I don’t know that there even is a permanent link for this that would be any good.

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.

• #### Joseph Nebus 6:00 pm on Sunday, 18 September, 2016 Permalink | Reply Tags: animation ( 8 ), arithmetic, calculus ( 84 ), notation ( 18 ), pizza ( 2 ), symbols ( 10 ), topology ( 10 )

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 of September, 2016. I don’t get to talking about the art of the comics here, but, I quite like Julian’s expressions here. And Bud Blake drew fantastic rumpled clothes.

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.

Jerry Scott and Jim Borgman’s Zits for the 14th of September, 2016. Properly speaking that is ink on his face, but I suppose saying it’s calculus pins down where it came from. Just observing.

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.

• #### ivasallay 3:44 am on Monday, 19 September, 2016 Permalink | Reply

You said, “Twelve depictions of apples are harder to read than the number 12.” It might be a little difficult to see at first, but the twelve apples were arranged to form the numerals 1 and 2. I thought it was rather clever.

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