My All 2020 Mathematics A to Z: Butterfly Effect


It’s a fun topic today, one suggested by Jacob Siehler, who I think is one of the people I met through Mathstodon. Mathstodon is a mathematics-themed instance of Mastodon, an open-source microblogging system. You can read its public messages here.

Color cartoon illustration of a coati in a beret and neckerchief, holding up a director's megaphone and looking over the Hollywood hills. The megaphone has the symbols + x (division obelus) and = on it. The Hollywood sign is, instead, the letters MATHEMATICS. In the background are spotlights, with several of them crossing so as to make the letters A and Z; one leg of the spotlights has 'TO' in it, so the art reads out, subtly, 'Mathematics A to Z'.
Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Butterfly Effect.

I take the short walk from my home to the Red Cedar River, and I pour a cup of water in. What happens next? To the water, anyway. Me, I think about walking all the way back home with this empty cup.

Let me have some simplifying assumptions. Pretend the cup of water remains somehow identifiable. That it doesn’t evaporate or dissolve into the riverbed. That it isn’t scooped up by a city or factory, drunk by an animal, or absorbed into a plant’s roots. That it doesn’t meet any interesting ions that turn it into other chemicals. It just goes as the river flows dictate. The Red Cedar River merges into the Grand River. This then moves west, emptying into Lake Michigan. Water from that eventually passes the Straits of Mackinac into Lake Huron. Through the St Clair River it goes to Lake Saint Clair, the Detroit River, Lake Erie, the Niagara River, the Niagara Falls, and Lake Ontario. Then into the Saint Lawrence River, then the Gulf of Saint Lawrence, before joining finally the North Atlantic.

Photograph of a small, tree-lined riverbed from a wooden bridge over it.
To the right: East Lansing and the Michigan State University campus. To the left, in a sense: the Atlantic Ocean.

If I pour in a second cup of water, somewhere else on the Red Cedar River, it has a similar journey. The details are different, but the course does not change. Grand River to Lake Michigan to three more Great Lakes to the Saint Lawrence to the North Atlantic Ocean. If I wish to know when my water passes the Mackinac Bridge I have a difficult problem. If I just wish to know what its future is, the problem is easy.

So now you understand dynamical systems. There’s some details to learn before you get a job, yes. But this is a perspective that explains what people in the field do, and why that. Dynamical systems are, largely, physics problems. They are about collections of things that interact according to some known potential energy. They may interact with each other. They may interact with the environment. We expect that where these things are changes in time. These changes are determined by the potential energies; there’s nothing random in it. Start a system from the same point twice and it will do the exact same thing twice.

We can describe the system as a set of coordinates. For a normal physics system the coordinates are the positions and momentums of everything that can move. If the potential energy’s rule changes with time, we probably have to include the time and the energy of the system as more coordinates. This collection of coordinates, describing the system at any moment, is a point. The point is somewhere inside phase space, which is an abstract idea, yes. But the geometry we know from the space we walk around in tells us things about phase space, too.

Imagine tracking my cup of water through its journey in the Red Cedar River. It draws out a thread, running from somewhere near my house into the Grand River and Lake Michigan and on. This great thin thread that I finally lose interest in when it flows into the Atlantic Ocean.

Dynamical systems drops in phase space act much the same. As the system changes in time, the coordinates of its parts change, or we expect them to. So “the point representing the system” moves. Where it moves depends on the potentials around it, the same way my cup of water moves according to the flow around it. “The point representing the system” traces out a thread, called a trajectory. The whole history of the system is somewhere on that thread.

Harvey, sulking: 'What a horrible year! How did it come to this?' Penny: 'I blame chaos theory.' Harvey: 'If it's chaos theory, I know EXACTLY who to blame! Some stupid butterfly flapped its wings and here we are.'
Stephen Beals’s Adult Children for the 21st of June, 2020. There were at least two “chaos theory”/“butterfly effect” comic strips in my feed just last week, and I wasn’t even looking. You can find comics essays where I talk about Adult Children at this link, and comic strips in general here.

Phase space, like a map, has regions. For my cup of water there’s a region that represents “is in Lake Michigan”. There’s another that represents “is going over Niagara Falls”. There’s one that represents “is stuck in Sandusky Bay a while”. When we study dynamical systems we are often interested in what these regions are, and what the boundaries between them are. Then a glance at where the point representing a system is tells us what it is doing. If the system represents a satellite orbiting a planet, we can tell whether it’s in a stable orbit, about to crash into a moon, or about to escape to interplanetary space. If the system represents weather, we can say it’s calm or stormy. If the system is a rigid pendulum — a favorite system to study, because we can draw its phase space on the blackboard — we can say whether the pendulum rocks back and forth or spins wildly.

Come back to my second cup of water, the one with a different history. It has a different thread from the first. So, too, a dynamical system started from a different point traces out a different trajectory. To find a trajectory is, normally, to solve differential equations. This is often useful to do. But from the dynamical systems perspective we’re usually interested in other issues.

For example: when I pour my cup of water in, does it stay together? The cup of water started all quite close together. But the different drops of water inside the cup? They’ve all had their own slightly different trajectories. So if I went with a bucket, one second later, trying to scoop it all up, likely I’d succeed. A minute later? … Possibly. An hour later? A day later?

By then I can’t gather it back up, practically speaking, because the water’s gotten all spread out across the Grand River. Possibly Lake Michigan. If I knew the flow of the river perfectly and knew well enough where I dropped the water in? I could predict where each goes, and catch each molecule of water right before it falls over Niagara. This is tedious but, after all, if you start from different spots — as the first and the last drop of my cup do — you expect to, eventually, go different places. They all end up in the North Atlantic anyway.

Photograph of Niagara Falls, showing the American Falls and the Bridal Veil, with a faint rainbow visible to the left of image and a boat sailing to the right.
Me, screaming to the pilot of the boat at center-right: “There’s my water drop! No, to the left! The left — your other left!”

Except … well, there is the Chicago Sanitary and Ship Canal. It connects the Chicago River to the Des Plaines River. The result is that some of Lake Michigan drains to the Ohio River, and from there the Mississippi River, and the Gulf of Mexico. There are also some canals in Ohio which connect Lake Erie to the Ohio River. I don’t know offhand of ones in Indiana or Wisconsin bringing Great Lakes water to the Mississippi. I assume there are, though.

Then, too, there is the Erie Canal, and the other canals of the New York State Canal System. These link the Niagara River and Lake Erie and Lake Ontario to the Hudson River. The Pennsylvania Canal System, too, links Lake Erie to the Delaware River. The Delaware and the Hudson may bring my water to the mid-Atlantic. I don’t know the canal systems of Ontario well enough to say whether some water goes to Hudson Bay; I’d grant that’s possible, though.

Think of my poor cups of water, now. I had been sure their fate was the North Atlantic. But if they happen to be in the right spot? They visit my old home off the Jersey Shore. Or they flow through Louisiana and warmer weather. What is their fate?

I will have butterflies in here soon.

Imagine two adjacent drops of water, one about to be pulled into the Chicago River and one with Lake Huron in its future. There is almost no difference in their current states. Their destinies are wildly separate, though. It’s surprising that so small a difference matters. Thinking through the surprise, it’s fair that this can happen, even for a deterministic system. It happens that there is a border, separating those bound for the Gulf and those for the North Atlantic, between these drops.

But how did those water drops get there? Where were they an hour before? … Somewhere else, yes. But still, on opposite sides of the border between “Gulf of Mexico water” and “North Atlantic water”. A day before, the drops were somewhere else yet, and the border was still between them. This separation goes back to, even, if the two drops came from my cup of water. Within the Red Cedar River is a border between a destiny of flowing past Quebec and of flowing past Saint Louis. And between flowing past Quebec and flowing past Syracuse. Between Syracuse and Philadelphia.

How far apart are those borders in the Red Cedar River? If you’ll go along with my assumptions, smaller than my cup of water. Not that I have the cup in a special location. The borders between all these fates are, probably, a complicated spaghetti-tangle. Anywhere along the river would be as fortunate. But what happens if the borders are separated by a space smaller than a drop? Well, a “drop” is a vague size. What if the borders are separated by a width smaller than a water molecule? There’s surely no subtleties in defining the “size” of a molecule.

That these borders are so close does not make the system random. It is still deterministic. Put a drop of water on this side of the border and it will go to this fate. But how do we know which side of the line the drop is on? If I toss this new cup out to the left rather than the right, does that matter? If my pinky twitches during the toss? If I am breathing in rather than out? What if a change too small to measure puts the drop on the other side?

And here we have the butterfly effect. It is about how a difference too small to observe has an effect too large to ignore. It is not about a system being random. It is about how we cannot know the system well enough for its predictability to tell us anything.

The term comes from the modern study of chaotic systems. One of the first topics in which the chaos was noticed, numerically, was weather simulations. The difference between a number’s representation in the computer’s memory and its rounded-off printout was noticeable. Edward Lorenz posed it aptly in 1963, saying that “one flap of a sea gull’s wings would be enough to alter the course of the weather forever”. Over the next few years this changed to a butterfly. In 1972 Philip Merrilees titled a talk Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? My impression is that these days the butterflies may be anywhere, and they alter hurricanes.

Comic strip Chaos Butterfly Man: 'Bitten by a radioactive chaos butterfly, Mike Mason gained the powers of a chaos butterfly!' [ Outside a bank ] Mason flaps his arm at an escaping robber. Robber: 'Ha-ha! What harm can THAT do me?' [ Nine days later ] Robber: 'Where did this thunderstorm finally come fr ... YOW!' (He's struck by lightning.)
RubenBolling’s Super-Fun-Pak Comix for the 23rd of May, 2020. Ruben Bolling uses Chaos Butterfly a good bit in both Super-Fun-Pak Comix and in the main strip, Tom the Dancing Bug. I have fewer essays exploring these Chaos Butterfly strips than you might imagine from that because I ran out of different things to say about the joke. Bolling’s is a great strip, though, and I recommend you consider it.

That we settle on butterflies as agents of chaos we can likely credit to their image. They seem to be innocent things so slight they barely exist. Hummingbirds probably move with too much obvious determination to fit the role. The Big Bad Wolf huffing and puffing would realistically be almost as nothing as a butterfly. But he has the power of myth to make him seem mightier than the storms. There are other happy accidents supporting butterflies, though. Edward Lorenz’s 1960s weather model makes trajectories that, plotted, create two great ellipsoids. The figures look like butterflies, all different but part of the same family. And there is Ray Bradbury’s classic short story, A Sound Of Thunder. If you don’t remember 7th grade English class, in the story time-travelling idiots change history, putting a fascist with terrible spelling in charge of a dystopian world, by stepping on a butterfly.

The butterfly then is metonymy for all the things too small to notice. Butterflies, sea gulls, turning the ceiling fan on in the wrong direction, prying open the living room window so there’s now a cross-breeze. They can matter, we learn.

Reading the Comics, March 11, 2020: Half Week Edition


There were a good number of comic strips mentioning mathematical subjects last week, as you might expect for one including the 14th of March. Most of them were casual mentions, though, so that’s why this essay looks like this. And is why the week will take two pieces to finish.

Jonathan Lemon and Joey Alison Sayer’s Little Oop for the 8th is part of a little storyline for the Sunday strips. In this the young Alley Oop has … travelled in time to the present. But different from how he does in the weekday strips. What’s relevant about this is Alley Oop hearing the year “2020” and mentioning how “we just got math where I come from” but being confident that’s either 40 or 400. Which itself follows up a little thread in the Sunday strips about new numbers on display and imagining numbers greater than three.

Venn Diagram with two bubbles. The left is 'Day after Daylight Savings [sic] Start'; the right is 'Monday'. The intersection has an arrow from it pointing to a travel cup of coffee.
Maria Scrivan’s Half Full for the 9th of March, 2020. Essays featuring some topic raised by Half Full appear at this link.

Maria Scrivan’s Half Full for the 9th is the Venn Diagram strip for the week.

Paul Trap’s Thatababy for the 9th is a memorial strip to Katherine Johnson. She was, as described, a NASA mathematician, and one of the great number of African-American women whose work computing was rescued from obscurity by the book and movie Hidden Figures. NASA, and its associated agencies, do a lot of mathematical work. Much of it is numerical mathematics: a great many orbital questions, for example, can not be answered with, like, the sort of formula that describes how far away a projectile launched on a parabolic curve will land. Creating a numerical version of a problem requires insight and thought about how to represent what we would like to know. And calculating that requires further insight, so that the calculation can be done accurately and speedily. (I think about sometime doing a bit about the sorts of numerical computing featured in the movie, but I would hardly be the first.)

Eulogy strip, as drawn by the baby, celebrating Katherine Johnson, NASA mathematician 1918 - 2020. It shows a child's drawing of her, and of a Mercury capsule, with formulas describing a ballistic trajectory making the motion trail of the capsule.
Paul Trap’s Thatababy for the 9th of March, 2020. My essays featuring something raised by Thatababy are at this link.

I also had thought the Mathematical Moments from the American Mathematical Society had posted an interview with her last year. I was mistaken but in, I think, a forgivable way. In the episode “Winning the Race”, posted the 12th of June, they interviewed Christine Darden, another of the people in the book, though not (really) the movie. Darden joined NASA in the late 60s. But the interview does talk about this sort of work, and how it evolved with technology. And, of course, mentions Johnson and her influence.

Graham Harrop’s Ten Cats for the 9th is another strip mentioning Albert Einstein and E = mc2. And using the blackboard full of symbols to represent deep thought.

Patrick Roberts’s Todd the Dinosaur for the 10th showcases Todd being terrified of fractions. And more terrified of story problems. I can’t call it a false representation of the kinds of mathematics that terrify people.

Teacher: 'All right, class, please take out your math books!' Todd: 'Teacher, this isn't gonna be fractions, is it?' Teacher: 'No, Todd, no fractions.' Todd: 'Whewwww!' Teacher: 'Now listen carefully, class. Train A leaves Chicago at 7:00 am, and ... ' (Todd, screaming in panic, runs out crashing through the wall and over the horizon.)
Patrick Roberts’s Todd the Dinosaur for the 10th of March, 2020. Essays that discuss something mentioned in a Todd the Dinosaur should be gathered at this link.

Stephen Beals’s Adult Children for the 11th has a character mourning that he took calculus as he’s “too stupid to be smart”. Knowing mathematics is often used as proof of intelligence. And calculus is used as the ultimate of mathematics. It’s a fair question why calculus and not some other field of mathematics, like differential equations or category theory or topology. Probably it’s a combination of slightly lucky choices (for calculus). Calculus is old enough to be respectable. It’s often taught as the ultimate mathematics course that people in high school or college (and who aren’t going into a mathematics field) will face. It’s a strange subject. Learning it requires a greater shift in thinking about how to solve problems than even learning algebra does. And the name is friendly enough, without the wordiness or technical-sounding language of, for example, differential equations. The subject may be well-situated.

Tony Rubino and Gary Markstein’s Daddy’s Home for the 11th has the pacing of a logic problem, something like the Liar’s Paradox. It’s also about homework which happens to be geometry, possibly because the cartoonists aren’t confident that kids that age might be taking a logic course.


I’ll have the rest of the week’s strips, including what Comic Strip Master Command ordered done for Pi Day, soon. And again I mention that I’m hosting this month’s Playful Math Education Blog Carnival. If you have come across a web site with some bit of mathematics that brought you delight and insight, please let me know, and mention any creative projects that you have, that I may mention that too. Thank you.

Reading the Comics, March 9, 2019: In Which I Explain Eleven Edition


I thought I had a flood of mathematically-themed comic strips last week. On reflection, many of them were slight enough not to need further context. You’ll see in the paragraph of not-discussed strips at the end of this. What did rate discussion turned out to get more interesting to me the more I wrote about them.

Stephen Beals’s Adult Children for the 6th uses mathematics as icon of things that are indisputably true. Two plus two equals four is a good example of such. If we take the ordinary meanings of ‘two’ and ‘plus’ and ‘equals’ and ‘four’ there’s no disputing it. The result follows from some uncontroversial-seeming axioms and a lot of deduction. By the rules of logic, the conclusion has to be true, whoever makes it. Even, for that matter, if nobody makes it. It’s difficult to imagine a universe in which nobody ever notices two plus two equals four. But we can imagine that there are mathematical truths that will never be noticed by anyone. (Here’s one. There is some largest finite whole number that any human-created project will ever use in any context. Consider the equation represented by “that number plus two equals (even bigger number)”.)

Harvey: 'Everyone ignores facts! Two plus two equals four, you know what I mean?' Friend: 'Yes. In your opinion, two plus two equals four.' Harvey: 'Noooo! Facts aren't opinions! There are no true facts, fake facts, iffy facts ... just facts! Let's judge things based on the facts!' Friend: 'And how do these facts make you feel?' Harvey, clutching his chest. 'Like you're giving me a fact attack.'
Stephen Beals’s Adult Children for the 6th of March, 2019. Essays inspired by something mentioned in Adult Children appear at this link.

But you see cards palmed there. What do we mean by ‘two’? Have we got a good definition? Might there be a different definition that’s more useful? Probably not, for ‘two’ anyway. But a part of mathematics, especially as a field develops, is working out what are the important concepts, and what their definitions should be. What a ‘function’ is, for example, went through a lot of debate and change over the 19th century. There is an elusiveness to facts, even in mathematics, where you’d think epistemology would be simpler.

Lauren's problem: '(x^2 y - 3y^2 + 5xy^2) - (-x^2 y + 3xy^2 - 3y^2). Which of the following is equivalent to the expression above? a. 4x^2 y^2. b. 8xy^2 - 6y^2. c. 2x^2 + 2xy^2. d. 2x^2 y + 8xy^2 - 6y^2.' Next problem: 'If a/b = 2 what's the value of 4b/a? a. 0. b. 1. c. 2. d. 4.' Bob, holding up empty ice trays: 'If a and b are empty because Lauren is selfish and not thinking of Bob, what are the chances he gets to have an iced drink? a. slim, b. none, c. all of the above?'
Frank Page’s Bob the Squirrel for the 6th of March, 2019. When I’m moved to write something based on Bob the Squirrel the essays should be tagged to appear at this link.

Frank Page’s Bob the Squirrel for the 6th continues the SAT prep questions from earlier in the week. There’s two more problems in shuffling around algebraic expressions here. The first one, problem 5, is probably easiest to do by eliminating wrong answers. (x^2 y - 3y^2 + 5xy^2) - (-x^2 y + 3xy^2 - 3y^2) is a tedious mess. But look at just the x^2 y terms: they have to add up to 2x^2 y , so, the answer has to be either c or d. So next look at the 3y^2 terms and oh, that’s nice. They add up to zero. The answer has to be c. If you feel like checking the 5xy^2 terms, go ahead; that’ll offer some reassurance, if you do the addition correctly.

The second one, problem 8, is probably easier to just think out. If \frac{a}{b} = 2 then there’s a lot of places to go. What stands out to me is that 4\frac{b}{a} has the reciprocal of \frac{a}{b} in it. So, the reciprocal of \frac{a}{b} has to equal the reciprocal of 2 . So \frac{a}{b} = \frac{1}{2} . And 4\frac{b}{a} is, well, four times \frac{b}{a} , so, four times one-half, or two. There’s other ways to go about this. In honestly, what I did when I looked at the problem was multiply both sides of \frac{a}{b} = 2 by \frac{b}{a} . But it’s harder to explain why that struck me as an obviously right thing to do. It’s got shortcuts I grew into from being comfortable with the more methodical approach. Someone who does a lot of problems like these will discover shortcuts.

Ruthie on the phone: 'Hello, homework hotline? I have an arithmetic question. Why isn't eleven called oneteen, and twelve called twoteen? ... You don't know? ... May I speak to your supervisor, please?'
Rick Detorie’s One Big Happy for the 6th of March, 2019. This particular strip is several years old, but I can’t pin down its original run more precisely than that. Essays featuring One Big Happy should be at this link.

Rick Detorie’s One Big Happy for the 6th asks one of those questions you need to be a genius or a child to ponder. Why don’t the numbers eleven and twelve follow the pattern of the other teens, or for that matter of twenty-one and thirty-two, and the like? And the short answer is that they kind of do. At least, “eleven” and “twelve”, etymologists agree, derive from the Proto-Germanic “ainlif” and “twalif”. If you squint your mouth you can get from “ain” to “one” (it’s probably easier if you go through the German “ein” along the way). Getting from “twa” to “two” is less hard. If my understanding is correct, etymologists aren’t fully agreed on the “lif” part. But they are settled on it means the part above ten. Like, “ainlif” would be “one left above ten”. So it parses as one-and-ten, putting it in form with the old London-English preference for one-and-twenty or two-and-thirty as word constructions.

It’s not hard to figure how “twalif” might over centuries mutate to “twelve”. We could ask why “thirteen” didn’t stay something more Old Germanic. My suspicion is that it amounts to just, well, it worked out like that. It worked out the same way in German, which switches to “-zehn” endings from 13 on. Lithuanian has all the teens end with “-lika”; Polish, similarly, but with “-ście”. Spanish — not a Germanic language — has “custom” words for the numbers up to 15, and then switches to “diecis-” as a prefix to the numbers 6 through 9. French doesn’t switch to a systematic pattern until 17. (And no I am not going to talk about France’s 80s and 90s.) My supposition is that different peoples came to different conclusions about whether they needed ten, or twelve, or fifteen, or sixteen, unique names for numbers before they had to resort to systemic names.

Here’s some more discussion of the teens, though, including some exploration of the controversy and links to other explanations.

Caption: '4 out of 5 Doctors agree ... ' Four, of five, chickens dressed as doctors: 'We are 80% of the doctors!'
Doug Savage’s Savage Chickens for the 6th of March, 2019. And the occasional essay based on Savage Chickens should be gathered at this link.

Doug Savage’s Savage Chickens for the 6th is a percentages comic. It makes reference to an old series of (American, at least) advertisements in which four out of five dentists would agree that chewing sugarless gum is a good thing. Shifting the four-out-of-five into 80% riffs is not just fun with tautologies. Percentages have this connotation of technical precision; 80% sounds like a more rigorously known number than “four out of five”. It doesn’t sound as scientific as “0.80”, quite. But when applied to populations a percentage seems less bizarre than a decimal.


Oh, now, and what about comic strips I can’t think of anything much to write about?
Ruben Bolling’s Super-Fun-Pak Comix for the 4th featured divisibility, in a panel titled “Fun Facts for the Obsessive-Compulsive”. Olivia James’s Nancy on the 6th was avoiding mathematics homework. Jonathan Mahood’s Bleeker: The Rechargeable Dog for the 7th has Skip avoiding studying for his mathematics test. Bob Scott’s Bear With Me for the 7th has Molly mourning a bad result on her mathematics test. (The comic strip was formerly known as Molly And The Bear, if this seems familiar but the name seems wrong.) These are all different comic strips, I swear. Bill Holbrook’s Kevin and Kell for the 8th has Rudy and Fiona in mathematics class. (The strip originally ran in 2013; Comics Kingdom has started running Holbrook’s web comic, but at several years’ remove.) And, finally, Alex Hallatt’s Human Cull for the 8th talks about “110%” as a phrase. I don’t mind the phrase, but the comic strip has a harder premise.


And that finishes the comic strips from last week. But Pi Day is coming. I’ll be ready for it. Shall see you there.

Reading the Comics, October 27, 2018: Surprise Rerun Edition


While putting together the last comics from a week ago I realized there was a repeat among them. And a pretty recent repeat too. I’m supposing this is a one-off, but who can be sure? We’ll get there. I figure to cover last week’s mathematically-themed comics in posts on Wednesday and Thursday, subject to circumstances.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 26th is a joking reminder that educational texts, including in mathematics, don’t have to be boring. We can have narrative thrust and energy. It’s a good reminder.

Caption: 'I wish all educational texts were written like Epictetus wrote.' Textbook: 'What, hapless wretch? Do you suppose int(sqrt(x^2 + x)dx = (x^2 + x)^{-1/2}? And when you eat, do you carry the food to your mouth or to your eyes? Slave!
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 26th of October, 2018.

As fits the joke, the bit of calculus in this textbook paragraph is wrong. \int \sqrt{x^2 + x} dx does not equal \left(x^2 + x\right)^{-\frac12} . This is even ignoring that we should expect, with an indefinite integral like this, a constant of integration. An indefinite integral like this is equal to a family of related functions. But it’s common shorthand to write out one representative function. But the indefinite integral of \sqrt{x^2 + x} is not \left(x^2 + x\right)^{-\frac12} . You can confirm that by differentiating \left(x^2 + x\right)^{-\frac12} . The result is nothing like \sqrt{x^2 + x} . Differentiating an indefinite integral should get the original function back. Here are the rules you need to do that for yourself.

As I make it out, a correct indefinite integral would be:

\int{\sqrt{x^2 + x} dx} = \frac{1}{4}\left( \left(2x + 1\right)\sqrt{x^2 + x} + \log \left|\sqrt{x} + \sqrt{x + 1} \right| \right)

Plus that “constant of integration” the value of which we can’t tell just from the function we want to indefinitely-integrate. I admit I haven’t double-checked that I’m right in my work here. I trust someone will tell me if I’m not. I’m going to feel proud enough if I can get the LaTeX there to display.

Berle: 'It was just a happy stroll through the gloomy graveyard when suddenly ... ' Spivak's Calculus, 3rd Edition appears. Berle: 'Math jumped out of nowhere!' Harvey: 'Drink it off.'
Stephen Beals’s Adult Children rerun for the 27th of October, 2018. It also appeared the 31st of March, 2018. I’m surprised it was that recent. I can’t blame Beals if he needed a break. This is a Halloween-ready example of the comic.

Stephen Beals’s Adult Children for the 27th has run already. It turned up in late March of this year. Michael Spivak’s Calculus is a good choice for representative textbook. Calculus holds its terrors, too. Even someone who’s gotten through trigonometry can find the subject full of weird, apparently arbitrary rules. And formulas like those in the above paragraph.

Manfred: 'How do you want to divide up the bill?' Wink: 'Lemme see it .. add that and that ... carry the two ... let's just split it htree ways.' Manfred: 'Nice try.' Dusty: 'Mr Surf 'n' Turf needs a calculator!'
Rob Harrell’s Big Top rerun for the 27th of October, 2018. It originally appeared the 13th of December, 2008.

Rob Harrell’s Big Top for the 27th is a strip about the difficulties of splitting a restaurant bill. And they’ve not even got to calculating the tip. (Maybe it’s just a strip about trying to push the group to splitting the bill a way that lets you off cheap. I haven’t had to face a group bill like this in several years. My skills with it are rusty.)

Jack-o-lantern standing on a scale: 'Hey! I weigh exactly 3.14 pounds!' Caption: 'Pumpkin Pi'.
Dave Whamond’s Reality Check for the 27th of October, 2018. Does the weight count if the jack-o-lantern is wearing sneakers?

Dave Whamond’s Reality Check for the 27th is a Pi Day joke shifted to the Halloween season.


And I have more Reading the Comics post at this link. Since it’s not true that every one of these includes a Saturday Morning Breakfast Cereal mention, you can find those that have one at this link. Essays discussing Adult Children, including the first time this particular strip appeared, are at this link. Essays with a mention of Big Top are at this link. And essays with a mention of Reality Check are at this link. Furthermore, this month and the rest of this year my Fall 2018 Mathematics A-To-Z should continue. And it is open for requests for more of the alphabet.

Reading the Comics, October 11, 2018: Under Weather Edition


I ended up not finding more comics on-topic on GoComics yesterday. So this past week’s mathematically-themed strips should fit into two posts well. I apologize for any loss of coherence in this essay, as I’m getting a bit of a cold. I’m looking forward to what this cold does for the A To Z essays coming Tuesday and Friday this week, too.

Stephen Beals’s Adult Children for the 7th uses Albert Einstein’s famous equation as shorthand for knowledge. I’m a little surprised it’s written out in words, rather than symbols. This might reflect that E = mc^2 is often understood just as this important series of sounds, rather than as an equation relating things to one another. Or it might just reflect the needs of the page composition. It could be too small a word balloon otherwise.

(In a darkened bar.) Harvey: 'Are they going to close?' Berle: 'They'll have to if this isn't fixed.' Harvey: 'E equals MC squared!' (The lights come on.) Berle 'Yay! ... Why did you yell that?' Harvey: 'Knowledge is power.'
Stephen Beals’s Adult Children for the 7th of October, 2018. I’m still not sure how I feel about this strip’s use of slightly offset panels like this.

Julie Larson’s The Dinette Set for the 9th continues the thread of tip-calculation jokes around here. I have no explanation for this phenomenon. In this case, Burl is doing the calculation correctly. If the tip is supposed to be 15% of the bill, and the bill is reduced 10%, then the tip would be reduced 10%. If you already have the tip calculated, it might be quicker to figure out a tenth of that rather than work out 15% of the original bill. And, yes, the characters are being rather unpleasantly penny-pinching. That was just the comic strip’s sense of humor.

Burl: 'So a 15% tip four two Monte Cristo platters would be $1.26' Dale: 'So ours would be the same as yours ... waidda minute! Today is 10% OFF for AARP members! So we times our total by 25% to figure the tip?' Burl: 'It's simple, Dale. Take 10% off the $1.26 tip, which is 12.6 cents, round that up to 13 cents, the minus that fro $1.26, and her tip is now $1.13.' Dale: 'Wow! I'm impressed! You did all that in your head! I'll bet I woulda given too much!'
Julie Larson’s The Dinette Set for the 9th of October, 2018. This is a rerun; it originally ran the 2nd of December, 2007.

Todd Clark’s Lola for the 9th take the form of your traditional grumbling about story problems. It also shows off the motif of updating of the words in a story problem to be awkwardly un-hip. The problem seems to be starting in a confounding direction anyway. The first sentence isn’t out and it’s introducing the rate at which Frank is shedding social-media friends over time and the rate at which a train is travelling, some distancer per time. Having one quantity with dimensions friends-per-time and another with dimensions distance-per-time is begging for confusion. Or for some weird gibberish thing, like, determining something to be (say) ninety mile-friends. There’s trouble ahead.

Lola: 'Sammy boy. How's middle school going?' Sammy: 'Stupid story problems.' Lola: 'Whatcha got? Maybe I can help.' Sammy: 'If Frank unfriends people at a rate of six per hour while on a train travelling ... '
Todd Clark’s Lola for the 9th of October, 2018. Don’t let your eye be distracted by the coloring job done on Lola’s eyeglasses in the last panel there.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 10th proposes naming a particular kind of series. A series is the sum of a sequence of numbers. It doesn’t have to be a sequence with infinitely many numbers in it, but it usually is, if it’s to be an interesting series. Properly, a series gets defined by something like the symbols in the upper caption of the panel:

\sum_{i = 1}^{\infty} a_i

Here the ‘i’ is a “dummy variable”, of no particular interest and not even detectable once the calculation is done. It’s not that thing with the square roots of -1 in thise case. ‘i’ is specifically known as the ‘index’, since it indexes the terms in the sequence. Despite the logic of i-index, I prefer to use ‘j’, ‘k’, or ‘n’. This avoids confusion with that square-root-of-minus-1 meaning for i. The index starts at some value, the one to the right of the equals sign underneath the capital sigma; in this case, 1. The sequence evaluates whatever the formula described by a_i is, for each whole number between that lowest ‘i’, in this case 1, and whatever the value above the sigma is. For the infinite series, that’s infinitely large. That is, work out a_i for every counting number ‘i’. For the first sum in the caption, that highest number is 4, and you only need to evaluate four terms and add them together. There’s no rule given for a_i in the caption; that just means that, in this case, we don’t yet have reason to care what the formula is.

On the blackboard: 'Solve: 24 + 12 + 6 + 3 + ... = ?' He put down 48. Woman: 'I ... wow. You've never studied series and you got it instantly.' Man: 'The 'plus three dots' part means 'plus 3', right?' Caption: 'New sequence type: Lucky Moron sequences. Definition: any convergent series such that 3 + (the first four terms) = (the infinite series)'.
Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 10th of October, 2018. And if you think that’s a not-really-needed name for a kind of series, note that MathWorld has a definition for the “FoxTrot Series” based on one problem from the FoxTrot comic strip from 1998.

This is the way to define a series if we’re being careful, and doing mathematics properly. But there are shorthands, and we fall back on them all the time. On the blackboard is one of them: 24 + 12 + 6 + 3 + \cdots . The \cdots at the end of a summation like this means “carry on this pattern for infinitely many terms”. If it appears in the middle of a summation, like 2 + 4 + 6 + 8 + \cdots + 20 it means “carry on this pattern for the appropriate number of terms”. In that case, it would be 10 + 12 + 14 + 16 + 18 .

The flaw with this “carry on this pattern” is that, properly, there’s no such thing as “the” pattern. There are infinitely many ways to continue from whatever the start was, and they’re all equally valid. What lets this scheme work is cultural expectations. We expect the difference between one term and the next to follow some easy patterns. They increase or decrease by the same amount as we’ve seen before (an arithmetic progression, like 2 + 4 + 6 + 8, increasing by two each time). They increase or decrease by the same ratio as we’ve seen before (a geometric progression, like 24 + 12 + 6 + 3, cutting in half each time). Maybe the sign alternates, or changes by some straightforward rule. If it isn’t one of these, then we have to fall back on being explicit. In this case, it would be that a_i = 24 \cdot \left(\frac{1}{2}\right)^{i - 1} .

The capital-sigma as shorthand for “sum” traces to Leonhard Euler, because of course. I’m finding it hard, in my copy of Florian Cajori’s History of Mathematical Notations, to find just where the series notation as we use it got started. Also I’m not finding where ellipses got into mathematical notation either. It might reflect everybody realizing this was a pretty good way to represent “we’re not going to write out the whole thing here”.

With something marked 30% off. First customer: 'This is normally 100 bucks. How much will it be at 30% off?' Val: '$70.' First customer, to second: 'See, I told you percents are the same as dollars.' Second customer: 'When you're right, you're right.' Val, thinking: 'I pity the next retailer who has to convince him otherwise.'
Norm Feuti’s Retail for the 11th of October, 2018. Yes, the comments include people explaining how people doing Common Core mathematics would never find an answer to “30% off $20”. Also a commenter who explains how one would, probably do it: “30% of 10 is 3. 20 is twice 20, so, 30% off 20 would be twice 3, or 6. So, 20 minus 6, or 14.” Followed by someone saying that if you did it by real math, it would be .7 times 20.

Norm Feuti’s Retail for the 11th riffs on how many people, fundamentally, don’t know what percentages are. I think it reflects thinking of a percentage as some kind of unit. We get used to measurements of things, like, pounds or seconds or dollars or degrees or such that are fixed in value. But a percentage is relative. It’s a fraction of some original quantity. A difference of (say) two pounds in weight is the same amount of weight whatever the original was; why wouldn’t two percent of the weight behave similarly? … Gads, yes, I feel for the next retailer who gets these customers.

I think I’ve already used the story from when I worked in the bookstore about the customer concerned whether the ten-percent-off sticker applied before or after sales tax was calculated. So I’ll only share if people ask to hear it. (They won’t ask.)


When I’m not getting a bit ill, I put my Reading the Comics posts at this link. Essays which mention Adult Children are at this link. Essays with The Dinette Set discussions should be at this link. The essays inspired by Lola are at this link. There’s some mention of Saturday Morning Breakfast Cereal in essays at link, or pretty much every Reading the Comics post. And Retail gets discussed at this link.

Reading the Comics, March 31, 2018: A Normal Week Edition


I have a couple loose rules about these Reading the Comics posts. At least one a week, whether there’s much to talk about or not. Not too many comics in one post, because that’s tiring to read and tiring to write. Trying to write up each day’s comics on the day mitigates that some, but not completely. So I tend to break up a week’s material if I can do, say, two posts of about seven strips each. This year, that’s been necessary; I’ve had a flood of comics on-topic or close enough for me to write about. This past week was a bizarre case. There really weren’t enough strips to break up the workload. It was, in short, a normal week, as strange as that is to see. I don’t know what I’m going to do Thursday. I might have to work.

Aaron McGruder’s Boondocks for the 25th of March is formally just a cameo mention of mathematics. There is some serious content to it. Whether someone likes to do a thing depends, to an extent, on whether they expect to like doing a thing. It seems likely to me that if a community encourages people to do mathematics, then it’ll have more people who do mathematics well. Mathematics does at least have the advantage that a lot of its fields can be turned into games. Or into things like games. Is one knot the same as another knot? You can test the laborious but inevitably correct way, trying to turn one into the other. Or you can find a polynomial that describes both knots and see if those two are the same polynomials. There’s fun to be had in this. I swear. And, of course, making arguments and finding flaws in other people’s arguments is a lot of mathematics. And good fun for anybody who likes that sort of thing. (This is a new tag for me.)

Huey: 'Ugh ... this video is terrible. Turn it.' Riley: 'You say everything is terrible. You're a hater.' Huey: 'Y'know ... the brutally honest critiques that you call 'hating' are why black people have always been at the forefront of music and culture. Artists knew that if they didn't excel, black people would yell and boo and heckle them off the stage Tough audiences have always made our artists better ... ' Riley: 'Uh-huh ... ' Huey: 'Now if we could only get black people to start booing each other in math class ... ' Riley: 'Whatever, hater.'
Aaron McGruder’s Boondocks for the 25th of March of March, 2018. Yeah, all right, but I would not want to be the teacher keeping a class of people heckling the student working a story problem on the board from turning abusive.

Ted Shearer’s Quincy for the 30th of January, 1979 and rerun the 26th names arithmetic as the homework Quincy’s most worried about. Or would like to put off the most. Harmless enough.

Quincy: 'I just heard one whole neighborhood is without electric power! I hope it's where my teacher lives.' Grandmother: 'Why?' Quincy: 'She won't be able to mark arithmetic papers.'
Ted Shearer’s Quincy for the 30th of January, 1979 and rerun the 26th of March, 2018. Because if there’s one thing that improves a teacher’s mood while grading, it’s having to do it while hurried after a night of rotten sleep in an apartment that possibly hasn’t got any heat!

Mike Thompson’s Grand Avenue for the 26th is a student-resisting-the-problem joke. A variable like ‘x’ serves a couple of roles. One of them is the name for a number whose value we don’t explicitly know, but which we hope to work out. And that’s the ‘x’ seen here. The other role of ‘x’ is the name for a number whose value we don’t know and don’t particularly care about. Since those are different reasons to use ‘x’ maybe we ought to have different names for the concepts. But we don’t and there’s probably no separating them now.

On the board: '17 + x = 21; solve for x'. Michael: 'x is unknown, so I'd hate to disrespect x's desire for privacy by disclosing its identity!'
Mike Thompson’s Grand Avenue for the 26th of March, 2018. And it took me more work than I wanted to figure out the kid’s name, so here: it’s Michael. Source: the Andrews-McMeel Syndication page about the comic. Cast page, since they use Javascript that keeps me from linking to that.

Tony Cochran’s Agnes for the 27th grumbles that mathematics and clairvoyance are poorly taught. Well, everyone who loves mathematics grumbles that the subject is poorly taught. I don’t know what the clairvoyants think but I’ll bet the same.

Agnes, thinking: 'In my mind, I'm seeing snow. Piles of it ... crippling traffic. Paralyzing the city. Scaring the fearless. Closing school.' Grandmom, clapping: 'Sun's out! School starts in an hour! Let's go! Hup! Hup! Hup' Agnes, thinking: 'Seems public schools excel at teaching clairvoyance as much as they do math.'
Tony Cochran’s Agnes for the 27th of March, 2018. In fairness, once students have got a little clairvoyant it becomes crazy hard to do assessment testing.

Mark Pett’s Lucky Cow rerun for the 28th is about sudoku. As with any puzzle the challenge is having rules that are restrictive enough to be interesting. This is also true of any mathematical field, though. You want ideas that imply a lot of things are true, but that also imply enough interesting plausible things are not true.

Leticia: 'You're doing a sudoku puzzle, Neil?' Neil: 'Yep! And I'm timing myself!' Leticia: 'How are you doing?' Neil: 'Really well, Leticia! See? I can complete it faster than the average!' Leticia: 'Wow! It's a challenge to fit the numbers in while following all the rules.' Neil, thinking: 'There are rules?'
Mark Pett’s Lucky Cow rerun for the 28th of March, 2018. It’s not that I’m not amused by the strip. But the mechanism of setting up the premise, developing it, and delivering the punch line really stands out sorely here. It’s hard to believe in someone saying Leticia’s line the third panel.

Rick DeTorie’s One Big Happy rerun for the 30th has Ruthie working on a story problem. One with loose change, which seems to turn up a lot in story problems. I never think of antes for some reason.

Dad: 'If Karen puts three quarters on the table ... and Kyle adds two nickels and one penny ... what would you have?' Ruthie: 'A very lopsided ante!'
Rick DeTorie’s One Big Happy rerun for the 30th of March, 2018. Grandpa plays a lot of card games with Ruthie maybe people should know.

Stephen Beals’s Adult Children for the 31st depicts mathematics as the stuff of nightmares. (Although it’s not clear to me this is meant to recount a nightmare. Reads like it, anyway.) Calculus, too, which is an interesting choice. Calculus seems to be a breaking point for many people. A lot of people even who were good at algebra or trigonometry find all this talk about differentials and integrals and limits won’t cohere into understanding. Isaac Asimov wrote about this several times, and the sad realization that for as much as he loved mathematics there were big important parts of it that he could not comprehend.

Berle: 'It was just a happy stroll through the gloomy graveyard when suddenly ... ' Spivak's Calculus, 3rd Edition appears. Berle: 'Math jumped out of nowhere!' Harvey: 'Drink it off.'
Stephen Beals’s Adult Children for the 31st of March, 2018. Spivak’s Calculus is one of the standard textbooks for intro students, by the way, although my own education was on James (not that James) Stewart. Spivak’s also noted for a well-regarded guide to TeX, which incidentally used a set of gender-neutral third-person singular pronouns (e[y]/em/eir) that some online communities embraced.

I’m curious why calculus should be such a discontinuity, but the reasons are probably straightforward. It’s a field where you’re less interested in doing things to numbers and more interested in doing things to functions. Or to curves that a function might represent. It’s a field where information about a whole region is important, rather than information about a single point. It’s a field where you can test your intuitive feeling for, say, a limit by calculating a couple of values, but for which those calculations don’t give the right answer. Or at least can’t be guaranteed to be right. I don’t know if the choice of what to represent mathematics was arbitrary. But it was a good choice certainly. (This is another newly-tagged strip.)