## Reading the Comics, September 8, 2017: First Split Week Edition, Part 1

It was looking like another slow week for something so early in the (United States) school year. Then Comic Strip Master Commend sent a flood of strips in for Friday and Saturday, so I’m splitting the load. It’s not a heavy one, as back-to-school jokes are on people’s minds. But here goes.

Marcus Hamilton and Scott Ketcham’s Dennis the Menace for the 3rd of September, 2017 is a fair strip for this early in the school year. It’s an old joke about making subtraction understandable.

Mark Anderson’s Andertoons for the 3rd is the Mark Anderson installment for this week, so I’m glad to have that. It’s a good old classic cranky-students setup and it reminds me that “unlike fractions” is a thing. I’m not quibbling with the term, especially not after the whole long-division mess a couple weeks back. I just hadn’t thought in a long while about how different denominators do make adding fractions harder.

Jeff Harris’s Shortcuts informational feature for the 3rd I couldn’t remember why I put on the list of mathematically-themed comic strips. The reason’s in there. There’s a Pi Joke. But my interest was more in learning that strawberries are a hybrid created in France from a North American and a Chilean breed. Isn’t that intriguing stuff?

Bill Abbott’s Specktickles for the 8th uses arithmetic — multiplication flash cards — as emblem of stuff to study. About all I can say for that.

## Reading the Comics, August 17, 2017: Professor Edition

To close out last week’s mathematically-themed comic strips … eh. There’s only a couple of them. One has a professor-y type and another has Albert Einstein. That’s enough for my subject line.

Joe Martin’s Mr Boffo for the 15th I’m not sure should be here. I think it’s a mathematics joke. That the professor’s shown with a pie chart suggests some kind of statistics, at least, and maybe the symbols are mathematical in focus. I don’t know. What the heck. I also don’t know how to link to these comics that gives attention to the comic strip artist. I like to link to the site from which I got the comic, but the Mr Boffo site is … let’s call it home-brewed. I can’t figure how to make it link to a particular archive page. But I feel bad enough losing Jumble. I don’t want to lose Joe Martin’s comics on top of that.

Charlie Podrebarac’s meat-and-Elvis-enthusiast comic Cow Town for the 15th is captioned “Elvis Disproves Relativity”. Of course it hasn’t anything to do with experimental results or even a good philosophical counterexample. It’s all about the famous equation. Have to expect that. Elvis Presley having an insight that challenges our understanding of why relativity should work is the stuff for sketch comedy, not single-panel daily comics.

Paul Trap’s Thatababy for the 15th has Thatadad win his fight with Alexa by using the old Star Trek Pi Gambit. To give a computer an unending task any number would work. Even the decimal digits of, say, five would do. They’d just be boring if written out in full, which is why we don’t. But irrational numbers at least give us a nice variety of digits. We don’t know that Pi is normal, but it probably is. So there should be a never-ending variety of what Alexa reels out here.

By the end of the strip Alexa has only got to the 55th digit of Pi after the decimal point. For this I use The Pi-Search Page, rather than working it out by myself. That’s what follows the digits in the second panel. So the comic isn’t skipping any time.

Gene Mora’s Graffiti for the 16th, if you count this as a comic strip, includes a pun, if you count this as a pun. Make of it what you like.

Mark Anderson’s Andertoons for the 17th is a student-misunderstanding-things problem. That’s a clumsy way to describe the joke. I should look for a punchier description, since there are a lot of mathematics comics that amount to the student getting a silly wrong idea of things. Well, I learned greater-than and less-than with alligators that eat the smaller number first. Though they turned into fish eating the smaller number first because who wants to ask a second-grade teacher to draw alligators all the time? Cartoon goldfish are so much easier.

## Reading the Comics, August 12, 2017: August 10 and 12 Edition

The other half of last week’s comic strips didn’t have any prominent pets in them. The six of them appeared on two days, though, so that’s as good as a particular theme. There’s also some π talk, but there’s enough of that I don’t want to overuse Pi Day as an edition name.

Mark Anderson’s Andertoons for the 10th is a classroom joke. It’s built on a common problem in teaching by examples. The student can make the wrong generalization. I like the joke. There’s probably no particular reason seven was used as the example number to have zero interact with. Maybe it just sounded funnier than the other numbers under ten that might be used.

Mike Baldwin’s Cornered for the 10th uses a chalkboard of symbols to imply deep thinking. The symbols on the board look to me like they’re drawn from some real mathematics or physics source. There’s force equations appropriate for gravity or electric interactions. I can’t explain the whole board, but that’s not essential to work out anyway.

Marty Links’s Emmy Lou for the 17th of March, 1976 was rerun the 10th of August. It name-drops the mathematics teacher as the scariest of the set. Fortunately, Emmy Lou went to her classes in a day before Rate My Professor was a thing, so her teacher doesn’t have to hear about this.

Scott Hilburn’s The Argyle Sweater for the 12th is a timely remidner that Scott Hilburn has way more Pi Day jokes than we have Pi Days to have. Also he has octopus jokes. It’s up to you to figure out whether the etymology of the caption makes sense.

John Zakour and Scott Roberts’s Working Daze for the 12th presents the “accountant can’t do arithmetic” joke. People who ought to be good at arithmetic being lousy at figuring tips is an ancient joke. I’m a touch surprised that Christopher Miller’s American Cornball: A Laffopedic Guide to the Formerly Funny doesn’t have an entry for tips (or mathematics). But that might reflect Miller’s mission to catalogue jokes that have fallen out of the popular lexicon, not merely that are old.

Michael Cavna’s Warped for the 12th is also a Pi Day joke that couldn’t wait. It’s cute and should fit on any mathematics teacher’s office door.

## Terrible and Less-Terrible Pi

As the 14th of March comes around it’s the time for mathematics bloggers to put up whatever they can about π. I will stir from my traditional crankiness about Pi Day (look, we don’t write days of the year as 3.14 unless we’re doing fake stardates) to bring back my two most π-relevant posts:

• Calculating Pi Terribly is about a probability-based way to calculate just what π’s digits are. It’s a lousy way to do it, but it works, technically.
• Calculating Pi Less Terribly is about an analysis-based way to calculate just what π’s digits are. It’s a less bad way to do it, although we actually use better-yet ways to work out the digits of a number like this.
• And what the heck, Normal Numbers, from an A To Z sequence. We do not actually know that π is a normal number. It’s the way I would bet, though, and here’s something about why I’d bet that way.

## Reading the Comics, March 4, 2017: Frazz, Christmas Trees, and Weddings Edition

It was another of those curious weeks when Comic Strip Master Command didn’t send quite enough comics my way. Among those they did send were a couple of strips in pairs. I can work with that.

Samson’s Dark Side Of The Horse for the 26th is the Roman Numerals joke for this essay. I apologize to Horace for being so late in writing about Roman Numerals but I did have to wait for Cecil Adams to publish first.

In Jef Mallett’s Frazz for the 26th Caulfield ponders what we know about Pythagoras. It’s hard to say much about the historical figure: he built a cult that sounds outright daft around himself. But it’s hard to say how much of their craziness was actually their craziness, how much was just that any ancient society had a lot of what seems nutty to us, and how much was jokes (or deliberate slander) directed against some weirdos. What does seem certain is that Pythagoras’s followers attributed many of their discoveries to him. And what’s certain is that the Pythagorean Theorem was known, at least a thing that could be used to measure things, long before Pythagoras was on the scene. I’m not sure if it was proved as a theorem or whether it was just known that making triangles with the right relative lengths meant you had a right triangle.

Greg Evans’s Luann Againn for the 28th of February — reprinting the strip from the same day in 1989 — uses a bit of arithmetic as generic homework. It’s an interesting change of pace that the mathematics homework is what keeps one from sleep. I don’t blame Luann or Puddles for not being very interested in this, though. Those sorts of complicated-fraction-manipulation problems, at least when I was in middle school, were always slogs of shuffling stuff around. They rarely got to anything we’d like to know.

Jef Mallett’s Frazz for the 1st of March is one of those little revelations that statistics can give one. Myself, I was always haunted by the line in Carl Sagan’s Cosmos about how, in the future, with the Sun ageing and (presumably) swelling in size and heat, the Earth would see one last perfect day. That there would most likely be quite fine days after that didn’t matter, and that different people might disagree on what made a day perfect didn’t matter. Setting out the idea of a “perfect day” and realizing there would someday be a last gave me chills. It still does.

Richard Thompson’s Poor Richard’s Almanac for the 1st and the 2nd of March have appeared here before. But I like the strip so I’ll reuse them too. They’re from the strip’s guide to types of Christmas trees. The Cubist Fur is described as “so asymmetrical it no longer inhabits Euclidean space”. Properly neither do we, but we can’t tell by eye the difference between our space and a Euclidean space. “Non-Euclidean” has picked up connotations of being so bizarre or even horrifying that we can’t hope to understand it. In practice, it means we have to go a little slower and think about, like, what would it look like if we drew a triangle on a ball instead of a sheet of paper. The Platonic Fir, in the 2nd of March strip, looks like a geometry diagram and I doubt that’s coincidental. It’s very hard to avoid thoughts of Platonic Ideals when one does any mathematics with a diagram. We know our drawings aren’t very good triangles or squares or circles especially. And three-dimensional shapes are worse, as see every ellipsoid ever done on a chalkboard. But we know what we mean by them. And then we can get into a good argument about what we mean by saying “this mathematical construct exists”.

Mark Litzler’s Joe Vanilla for the 3rd uses a chalkboard full of mathematics to represent the deep thinking behind a silly little thing. I can’t make any of the symbols out to mean anything specific, but I do like the way it looks. It’s quite well-done in looking like the shorthand that, especially, physicists would use while roughing out a problem. That there are subscripts with forms like “12” and “22” with a bar over them reinforces that. I would, knowing nothing else, expect this to represent some interaction between particles 1 and 2, and 2 with itself, and that the bar means some kind of complement. This doesn’t mean much to me, but with luck, it means enough to the scientist working it out that it could be turned into a coherent paper.

Bill Holbrook’s On The Fastrack is this week about the wedding of the accounting-minded Fi. And she’s having last-minute doubts, which is why the strip of the 3rd brings in irrational and anthropomorphized numerals. π gets called in to serve as emblematic of the irrational numbers. Can’t fault that. I think the only more famously irrational number is the square root of two, and π anthropomorphizes more easily. Well, you can draw an established character’s face onto π. The square root of 2 is, necessarily, at least two disconnected symbols and you don’t want to raise distracting questions about whether the root sign or the 2 gets the face.

That said, it’s a lot easier to prove that the square root of 2 is irrational. Even the Pythagoreans knew it, and a bright child can follow the proof. A really bright child could create a proof of it. To prove that π is irrational is not at all easy; it took mathematicians until the 19th century. And the best proof I know of the fact does it by a roundabout method. We prove that if a number (other than zero) is rational then the tangent of that number must be irrational, and vice-versa. And the tangent of π/4 is 1, so therefore π/4 must be irrational, so therefore π must be irrational. I know you’ll all trust me on that argument, but I wouldn’t want to sell it to a bright child.

Holbrook continues the thread on the 4th, extends the anthropomorphic-mathematics-stuff to call people variables. There’s ways that this is fair. We use a variable for a number whose value we don’t know or don’t care about. A “random variable” is one that could take on any of a set of values. We don’t know which one it does, in any particular case. But we do know — or we can find out — how likely each of the possible values is. We can use this to understand the behavior of systems even if we never actually know what any one of it does. You see how I’m going to defend this metaphor, then, especially if we allow that what people are likely or unlikely to do will depend on context and evolve in time.

## Reading the Comics, February 15, 2017: SMBC Cuts In Line Edition

It’s another busy enough week for mathematically-themed comic strips that I’m dividing the harvest in two. There’s a natural cutting point since there weren’t any comics I could call relevant for the 15th. But I’m moving a Saturday Morning Breakfast Cereal of course from the 16th into this pile. That’s because there’s another Saturday Morning Breakfast Cereal of course from after the 16th that I might include. I’m still deciding if it’s close enough to on topic. We’ll see.

John Graziano’s Ripley’s Believe It Or Not for the 12th mentions the “Futurama Theorem”. The trivia is true, in that writer Ken Keeler did create a theorem for a body-swap plot he had going. The premise was that any two bodies could swap minds at most one time. So, after a couple people had swapped bodies, was there any way to get everyone back to their correct original body? There is, if you bring two more people in to the body-swapping party. It’s clever.

From reading comment threads about the episode I conclude people are really awestruck by the idea of creating a theorem for a TV show episode. The thing is that “a theorem” isn’t necessarily a mind-boggling piece of work. It’s just the name mathematicians give when we have a clearly-defined logical problem and its solution. A theorem and its proof can be a mind-wrenching bit of work, like Fermat’s Last Theorem or the Four-Color Map Theorem are. Or it can be on the verge of obvious. Keeler’s proof isn’t on the obvious side of things. But it is the reasoning one would have to do to solve the body-swap problem the episode posited without cheating. Logic and good story-telling are, as often, good partners.

Teresa Burritt’s Frog Applause is a Dadaist nonsense strip. But for the 13th it hit across some legitimate words, about a 14 percent false-positive rate. This is something run across in hypothesis testing. The hypothesis is something like “is whatever we’re measuring so much above (or so far below) the average that it’s not plausibly just luck?” A false positive is what it sounds like: our analysis said yes, this can’t just be luck, and it turns out that it was. This turns up most notoriously in medical screenings, when we want to know if there’s reason to suspect a health risk, and in forensic analysis, when we want to know if a particular person can be shown to have been a particular place at a particular time. A 14 percent false positive rate doesn’t sound very good — except.

Suppose we are looking for a rare condition. Say, something one person out of 500 will have. A test that’s 99 percent accurate will turn up positives for the one person who has got it and for five of the people who haven’t. It’s not that the test is bad; it’s just there are so many negatives to work through. If you can screen out a good number of the negatives, though, the people who haven’t got the condition, then the good test will turn up fewer false positives. So suppose you have a cheap or easy or quick test that doesn’t miss any true positives but does have a 14 percent false positive rate. That would screen out 430 of the people who haven’t got whatever we’re testing for, leaving only 71 people who need the 99-percent-accurate test. This can make for a more effective use of resources.

Gary Wise and Lance Aldrich’s Real Life Adventures for the 13th is an algebra-in-real-life joke and I can’t make something deeper out of that.

Mike Shiell’s The Wandering Melon for the 13th is a spot of wordplay built around statisticians. Good for taping to the mathematics teacher’s walls.

Eric the Circle for the 14th, this one by “zapaway”, is another bit of wordplay. Tans and tangents.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 16th identifies, aptly, a difference between scientists and science fans. Weinersmith is right that loving trivia is a hallmark of a fan. Expertise — in any field, not just science — is more about recognizing patterns of problems and concepts, ways to bring approaches from one field into another, this sort of thing. And the digits of π are great examples of trivia. There’s no need for anyone to know the 1,681st digit of π. There’s few calculations you could ever do when you needed more than three dozen digits. But if memorizing digits seems like fun then π is a great set to learn. e is the only other number at all compelling.

The thing is, it’s very hard to become an expert in something without first being a fan of it. It’s possible, but if a field doesn’t delight you why would you put that much work into it? So even though the scientist might have long since gotten past caring how many digits of π, it’s awfully hard to get something memorized in the flush of fandom out of your head.

I know you’re curious. I can only remember π out to 3.14158926535787962. I might have gotten farther if I’d tried, but I actually got a digit wrong, inserting a ‘3’ before that last ’62’, and the effort to get that mistake out of my head obliterated any desire to waste more time memorizing digits. For e I can only give you 2.718281828. But there’s almost no hope I’d know that far if it weren’t for how e happens to repeat that 1828 stanza right away.

## The End 2016 Mathematics A To Z: Normal Numbers

Today’s A To Z term is another of gaurish’s requests. It’s also a fun one so I’m glad to have reason to write about it.

## Normal Numbers

A normal number is any real number you never heard of.

Yeah, that’s not what we say a normal number is. But that’s what a normal number is. If we could imagine the real numbers to be a stream, and that we could reach into it and pluck out a water-drop that was a single number, we know what we would likely pick. It would be an irrational number. It would be a transcendental number. And it would be a normal number.

We know normal numbers — or we would, anyway — by looking at their representation in digits. For example, π is a number that starts out 3.1415926535897931159979634685441851615905 and so on forever. Look at those digits. Some of them are 1’s. How many? How many are 2’s? How many are 3’s? Are there more than you would expect? Are there fewer? What would you expect?

Expect. That’s the key. What should we expect in the digits of any number? The numbers we work with don’t offer much help. A whole number, like 2? That has a decimal representation of a single ‘2’ and infinitely many zeroes past the decimal point. Two and a half? A single ‘2, a single ‘5’, and then infinitely many zeroes past the decimal point. One-seventh? Well, we get infinitely many 1’s, 4’s, 2’s, 8’s, 5’s, and 7’s. Never any 3’s, nor any 0’s, nor 6’s or 9’s. This doesn’t tell us anything about how often we would expect ‘8’ to appear in the digits of π.

In an normal number we get all the decimal digits. And we get each of them about one-tenth of the time. If all we had was a chart of how often digits turn up we couldn’t tell the summary of one normal number from the summary of any other normal number. Nor could we tell either from the summary of a perfectly uniform randomly drawn number.

It goes beyond single digits, though. Look at pairs of digits. How often does ’14’ turn up in the digits of a normal number? … Well, something like once for every hundred pairs of digits you draw from the random number. Look at triplets of digits. ‘141’ should turn up about once in every thousand sets of three digits. ‘1415’ should turn up about once in every ten thousand sets of four digits. Any finite string of digits should turn up, and exactly as often as any other finite string of digits.

That’s in the full representation. If you look at all the infinitely many digits the normal number has to offer. If all you have is a slice then some digits are going to be more common and some less common. That’s similar to how if you fairly toss a coin (say) forty times, there’s a good chance you’ll get tails something other than exactly twenty times. Look at the first 35 or so digits of π there’s not a zero to be found. But as you survey more digits you get closer and closer to the expected average frequency. It’s the same way coin flips get closer and closer to 50 percent tails. Zero is a rarity in the first 35 digits. It’s about one-tenth of the first 3500 digits.

The digits of a specific number are not random, not if we know what the number is. But we can be presented with a subset of its digits and have no good way of guessing what the next digit might be. That is getting into the same strange territory in which we can speak about the “chance” of a month having a Friday the 13th even though the appearances of Fridays the 13th have absolutely no randomness to them.

This has staggering implications. Some of them inspire an argument in science fiction Usenet newsgroup rec.arts.sf.written every two years or so. Probably it does so in other venues; Usenet is just my first home and love for this. In a minor point in Carl Sagan’s novel Cosmos possibly-imaginary aliens reveal there’s a pattern hidden in the digits of π. (It’s not in the movie version, which is a shame. But to include it would require people watching a computer. So that could not make for a good movie scene, we now know.) Look far enough into π, says the book, and there’s suddenly a string of digits that are nearly all zeroes, interrupted with a few ones. Arrange the zeroes and ones into a rectangle and it draws a pixel-art circle. And the aliens don’t know how something astounding like that could be.

Nonsense, respond the kind of science fiction reader that likes to identify what the nonsense in science fiction stories is. (Spoiler: it’s the science. In this case, the mathematics too.) In a normal number every finite string of digits appears. It would be truly astounding if there weren’t an encoded circle in the digits of π. Indeed, it would be impossible for there not to be infinitely many circles of every possible size encoded in every possible way in the digits of π. If the aliens are amazed by that they would be amazed to find how every triangle has three corners.

I’m a more forgiving reader. And I’ll give Sagan this amazingness. I have two reasons. The first reason is on the grounds of discoverability. Yes, the digits of a normal number will have in them every possible finite “message” encoded every possible way. (I put the quotes around “message” because it feels like an abuse to call something a message if it has no sender. But it’s hard to not see as a “message” something that seems to mean something, since we live in an era that accepts the Death of the Author as a concept at least.) Pick your classic cypher `1 = A, 2 = B, 3 = C’ and so on, and take any normal number. If you look far enough into its digits you will find every message you might ever wish to send, every book you could read. Every normal number holds Jorge Luis Borges’s Library of Babel, and almost every real number is a normal number.

But. The key there is if you look far enough. Look above; the first 35 or so digits of π have no 0’s, when you would expect three or four of them. There’s no 22’s, even though that number has as much right to appear as does 15, which gets in at least twice that I see. And we will only ever know finitely many digits of π. It may be staggeringly many digits, sure. It already is. But it will never be enough to be confident that a circle, or any other long enough “message”, must appear. It is staggering that a detectable “message” that long should be in the tiny slice of digits that we might ever get to see.

And it’s harder than that. Sagan’s book says the circle appears in whatever base π gets represented in. So not only does the aliens’ circle pop up in base ten, but also in base two and base sixteen and all the other, even less important bases. The circle happening to appear in the accessible digits of π might be an imaginable coincidence in some base. There’s infinitely many bases, one of them has to be lucky, right? But to appear in the accessible digits of π in every one of them? That’s staggeringly impossible. I say the aliens are correct to be amazed.

Now to my second reason to side with the book. It’s true that any normal number will have any “message” contained in it. So who says that π is a normal number?

We think it is. It looks like a normal number. We have figured out many, many digits of π and they’re distributed the way we would expect from a normal number. And we know that nearly all real numbers are normal numbers. If I had to put money on it I would bet π is normal. It’s the clearly safe bet. But nobody has ever proved that it is, nor that it isn’t. Whether π is normal or not is a fit subject for conjecture. A writer of science fiction may suppose anything she likes about its normality without current knowledge saying she’s wrong.

It’s easy to imagine numbers that aren’t normal. Rational numbers aren’t, for example. If you followed my instructions and made your own transcendental number then you made a non-normal number. It’s possible that π should be non-normal. The first thirty million digits or so look good, though, if you think normal is good. But what’s thirty million against infinitely many possible counterexamples? For all we know, there comes a time when π runs out of interesting-looking digits and turns into an unpredictable little fluttering between 6 and 8.

It’s hard to prove that any numbers we’d like to know about are normal. We don’t know about π. We don’t know about e, the base of the natural logarithm. We don’t know about the natural logarithm of 2. There is a proof that the square root of two (and other non-square whole numbers, like 3 or 5) is normal in base two. But my understanding is it’s a nonstandard approach that isn’t quite satisfactory to experts in the field. I’m not expert so I can’t say why it isn’t quite satisfactory. If the proof’s authors or grad students wish to quarrel with my characterization I’m happy to give space for their rebuttal.

It’s much the way transcendental numbers were in the 19th century. We understand there to be this class of numbers that comprises nearly every number. We just don’t have many examples. But we’re still short on examples of transcendental numbers. Maybe we’re not that badly off with normal numbers.

We can construct normal numbers. For example, there’s the Champernowne Constant. It’s the number you would make if you wanted to show you could make a normal number. It’s 0.12345678910111213141516171819202122232425 and I bet you can imagine how that develops from that point. (David Gawen Champernowne proved it was normal, which is the hard part.) There’s other ways to build normal numbers too, if you like. But those numbers aren’t of any interest except that we know them to be normal.

Mere normality is tied to a base. A number might be normal in base ten (the way normal people write numbers) but not in base two or base sixteen (which computers and people working on computers use). It might be normal in base twelve, used by nobody except mathematics popularizers of the 1960s explaining bases, but not normal in base ten. There can be numbers normal in every base. They’re called “absolutely normal”. Nearly all real numbers are absolutely normal. Wacław Sierpiński constructed the first known absolutely normal number in 1917. If you got in on the fractals boom of the 80s and 90s you know his name, although without the Polish spelling. He did stuff with gaskets and curves and carpets you wouldn’t believe. I’ve never seen Sierpiński’s construction of an absolutely normal number. From my references I’m not sure if we know how to construct any other absolutely normal numbers.

So that is the strange state of things. Nearly every real number is normal. Nearly every number is absolutely normal. We know a couple normal numbers. We know at least one absolutely normal number. But we haven’t (to my knowledge) proved any number that’s otherwise interesting is also a normal number. This is why I say: a normal number is any real number you never heard of.

## Reading the Comics, October 19, 2016: An Extra Day Edition

I didn’t make noise about it, but last Sunday’s mathematics comic strip roundup was short one day. I was away from home and normal computer stuff Saturday. So I posted without that day’s strips under review. There was just the one, anyway.

Also I want to remind folks I’m doing another Mathematics A To Z, and taking requests for words to explain. There are many appealing letters still unclaimed, including ‘A’, ‘T’, and ‘O’. Please put requests in over on that page because. It’s easier for me to keep track of what’s been claimed that way.

Matt Janz’s Out of the Gene Pool rerun for the 15th missed last week’s cut. It does mention the Law of Cosines, which is what the Pythagorean Theorem looks like if you don’t have a right triangle. You still have to have a triangle. Bobby-Sue recites the formula correctly, if you know the notation. The formula’s $c^2 = a^2 + b^2 - 2 a b \cos\left(C\right)$. Here ‘a’ and ‘b’ and ‘c’ are the lengths of legs of the triangle. ‘C’, the capital letter, is the size of the angle opposite the leg with length ‘c’. That’s a common notation. ‘A’ would be the size of the angle opposite the leg with length ‘a’. ‘B’ is the size of the angle opposite the leg with length ‘b’. The Law of Cosines is a generalization of the Pythagorean Theorem. It’s a result that tells us something like the original theorem but for cases the original theorem can’t cover. And if it happens to be a right triangle the Law of Cosines gives us back the original Pythagorean Theorem. In a right triangle C is the size of a right angle, and the cosine of that is 0.

That said Bobby-Sue is being fussy about the drawings. No geometrical drawing is ever perfectly right. The universe isn’t precise enough to let us draw a right triangle. Come to it we can’t even draw a triangle, not really. We’re meant to use these drawings to help us imagine the true, Platonic ideal, figure. We don’t always get there. Mock proofs, the kind of geometric puzzle showing something we know to be nonsense, rely on that. Give chalkboard art a break.

Samson’s Dark Side of the Horse for the 17th is the return of Horace-counting-sheep jokes. So we get a π joke. I’m amused, although I couldn’t sleep trying to remember digits of π out quite that far. I do better working out Collatz sequences.

Hilary Price’s Rhymes With Orange for the 19th at least shows the attempt to relieve mathematics anxiety. I’m sympathetic. It does seem like there should be ways to relieve this (or any other) anxiety, but finding which ones work, and which ones work best, is partly a mathematical problem. As often happens with Price’s comics I’m particularly tickled by the gag in the title panel.

Norm Feuti’s Gil rerun for the 19th builds on the idea calculators are inherently cheating on arithmetic homework. I’m sympathetic to both sides here. If Gil just wants to know that his answers are right there’s not much reason not to use a calculator. But if Gil wants to know that he followed the right process then the calculator’s useless. By the right process I mean, well, the work to be done. Did he start out trying to calculate the right thing? Did he pick an appropriate process? Did he carry out all the steps in that process correctly? If he made mistakes on any of those he probably didn’t get to the right answer, but it’s not impossible that he would. Sometimes multiple errors conspire and cancel one another out. That may not hurt you with any one answer, but it does mean you aren’t doing the problem right and a future problem might not be so lucky.

Zach Weinersmith’s Saturday Morning Breakfast Cereal rerun for the 19th has God crashing a mathematics course to proclaim there’s a largest number. We can suppose there is such a thing. That’s how arithmetic modulo a number is done, for one. It can produce weird results in which stuff we just naturally rely on doesn’t work anymore. For example, in ordinary arithmetic we know that if one number times another equals zero, then either the first number or the second, or both, were zero. We use this in solving polynomials all the time. But in arithmetic modulo 8 (say), 4 times 2 is equal to 0.

And if we recklessly talk about “infinity” as a number then we get outright crazy results, some of them teased in Weinersmith’s comic. “Infinity plus one”, for example, is “infinity”. So is “infinity minus one”. If we do it right, “infinity minus infinity” is “infinity”, or maybe zero, or really any number you want. We can avoid these logical disasters — so far, anyway — by being careful. We have to understand that “infinity” is not a number, though we can use numbers growing infinitely large.

Induction, meanwhile, is a great, powerful, yet baffling form of proof. When it solves a problem it solves it beautifully. And easily, too, usually by doing something like testing two special cases. Maybe three. At least a couple special cases of whatever you want to know. But picking the cases, and setting them up so that the proof is valid, is not easy. There’s logical pitfalls and it is so hard to learn how to avoid them.

Jon Rosenberg’s Scenes from a Multiverse for the 19th plays on a wonderful paradox of randomness. Randomness is … well, unpredictable. If I tried to sell you a sequence of random numbers and they were ‘1, 2, 3, 4, 5, 6, 7’ you’d be suspicious at least. And yet, perfect randomness will sometimes produce patterns. If there were no little patches of order we’d have reason to suspect the randomness was faked. There is no reason that a message like “this monkey evolved naturally” couldn’t be encoded into a genome by chance. It may just be so unlikely we don’t buy it. The longer the patch of order the less likely it is. And yet, incredibly unlikely things do happen. The study of impossibly unlikely events is a good way to quickly break your brain, in case you need one.

## Reading the Comics, September 6, 2016: Oh Thank Goodness We’re Back Edition

That’s a relief. After the previous week’s suspicious silence Comic Strip Master Command sent a healthy number of mathematically-themed comics my way. They cover a pretty normal spread of topics. So this makes for a nice normal sort of roundup.

Mac King and Bill King’s Magic In A Minute for the 4th is an arithmetic-magic-trick. Like most arithmetic-magic it depends on some true but, to me, dull bit of mathematics. In this case, that 81,234,567 minus 12,345,678 is equal to something. As a kid this sort of trick never impressed me because, well, anyone can do subtraction. I didn’t appreciate that the fun of stage magic in presenting well the mundane.

Jerry Scott and Jim Borgman’s Zits for the 5th is an ordinary mathematics-is-hard joke. But it’s elevated by the artwork, which shows off the expressive and slightly surreal style that makes the comic so reliable and popular. The formulas look fair enough, the sorts of things someone might’ve been cramming before class. If they’re a bit jumbled up, well, Pierce hasn’t been well.

Jeffrey Caulfield and Alexandre Rouillard’s Mustard and Boloney for the 6th is an anthropomorphic-shapes joke and I feel like it’s been here before. Ah, yeah, there it is, from about this time last year. It’s a fair one to rerun.

Mustard and Boloney popped back in on the 8th with a strip I don’t have in my archive at least. It’s your standard Pi Pun, though. If they’re smart they’ll rerun it in March. I like the coloring; it’s at least a pleasant panel to look at.

Percy Crosby’s Skippy from the 9th of July, 1929 was rerun the 6th of September. It seems like a simple kid-saying-silly-stuff strip: what is the difference between the phone numbers Clinton 2651 and Clinton 2741 when they add to the same number? (And if Central knows what the number is why do they waste Skippy’s time correcting him? And why, 87 years later, does the phone yell at me for not guessing correctly whether I need the area code for a local number and whether I need to dial 1 before that?) But then who cares what the digits in a telephone number add to? What could that tell us about anything?

As phone numbers historically developed, the sum can’t tell us anything at all. But if we had designed telephone numbers correctly we could have made it … not impossible to dial a wrong number, but at least made it harder. This insight comes to us from information theory, which, to be fair, we have because telephone companies spent decades trying to work out solutions to problems like people dialing numbers wrong or signals getting garbled in the transmission. We can allow for error detection by schemes as simple as passing along, besides the numbers, the sum of the numbers. This can allow for the detection of a single error: had Skippy called for number 2641 instead of 2741 the problem would be known. But it’s helpless against two errors, calling for 2541 instead of 2741. But we could detect a second error by calculating some second term based on the number we wanted, and sending that along too.

By adding some more information, other modified sums of the digits we want, we can even start correcting errors. We understand the logic of this intuitively. When we repeat a message twice after sending it, we are trusting that even if one copy of the message is garbled the recipient will take the version received twice as more likely what’s meant. We can design subtler schemes, ones that don’t require we repeat the number three times over. But that should convince you that we can do it.

The tradeoff is obvious. We have to say more digits of the number we want. It isn’t hard to reach the point we’re ending more error-detecting and error-correcting numbers than we are numbers we want. And what if we make a mistake in the error-correcting numbers? (If we used a smart enough scheme, we can work out the error was in the error-correcting number, and relax.) If it’s important that we get the message through, we shrug and accept this. If there’s no real harm done in getting the message wrong — if we can shrug off the problem of accidentally getting the wrong phone number — then we don’t worry about making a mistake.

And at this point we’re only a few days into the week. I have enough hundreds of words on the close of the week I’ll put off posting that a couple of days. It’s quite good having the comics back to normal.

## Reading the Comics, June 26, 2015: June 23, 2016 Plus Golden Lizards Edition

And now for the huge pile of comic strips that had some mathematics-related content on the 23rd of June. I admit some of them are just using mathematics as a stand-in for “something really smart people do”. But first, another moment with the Magic Realism Bot:

So, you know, watch the lizards and all.

Tom Batiuk’s Funky Winkerbean name-drops E = mc2 as the sort of thing people respect. If the strip seems a little baffling then you should know that Mason’s last name is Jarr. He was originally introduced as a minor player in a storyline that wasn’t about him, so the name just had to exist. But since then Tom Batiuk’s decided he likes the fellow and promoted him to major-player status. And maybe Batiuk regrets having a major character with a self-consciously Funny Name, which is an odd thing considering he named his long-running comic strip for original lead character Funky Winkerbean.

Charlie Podrebarac’s CowTown depicts the harsh realities of Math Camp. I assume they’re the realities. I never went to one myself. And while I was on the Physics Team in high school I didn’t make it over to the competitive mathematics squad. Yes, I noticed that the not-a-numbers-person Jim Smith can’t come up with anything other than the null symbol, representing nothing, not even zero. I like that touch.

Ryan North’s Dinosaur Comics rerun is about Richard Feynman, the great physicist whose classic memoir What Do You Care What Other People Think? is hundreds of pages of stories about how awesome he was. Anyway, the story goes that Feynman noticed one of the sequences of digits in π and thought of the joke which T-Rex shares here.

π is believed but not proved to be a “normal” number. This means several things. One is that any finite sequence of digits you like should appear in its representation, somewhere. Feynman and T-Rex look for the sequence ‘999999’, which sure enough happens less than eight hundred digits past the decimal point. Lucky stroke there. There’s no reason to suppose the sequence should be anywhere near the decimal point. There’s no reason to suppose the sequence has to be anywhere in the finite number of digits of π that humanity will ever know. (This is why Carl Sagan’s novel Contact, which has as a plot point the discovery of a message apparently encoded in the digits of π, is not building on a stupid idea. That any finite message exists somewhere is kind-of certain. That it’s findable is not.)

e, mentioned in the last panel, is similarly thought to be a normal number. It’s also not proved to be. We are able to say that nearly all numbers are normal. It’s in much the way we can say nearly all numbers are irrational. But it is hard to prove that any numbers are. I believe that the only numbers humans have proved to be normal are a handful of freaks created to show normal numbers exist. I don’t know of any number that’s interesting in its own right that’s also been shown to be normal. We just know that almost all numbers are.

But it is imaginable that π or e aren’t. They look like they’re normal, based on how their digits are arranged. It’s an open question and someone might make a name for herself by answering the question. It’s not an easy question, though.

Missy Meyer’s Holiday Doodles breaks the news to me the 23rd was SAT Math Day. I had no idea and I’m not sure what that even means. The doodle does use the classic “two trains leave Chicago” introduction, the “it was a dark and stormy night” of Boring High School Algebra word problems.

Stephan Pastis’s Pearls Before Swine is about everyone who does science and mathematics popularization, and what we worry someone’s going to reveal about us. Um. Except me, of course. I don’t do this at all.

Ashleigh Brilliant’s Pot-Shots rerun is a nice little averages joke. It does highlight something which looks paradoxical, though. Typically if you look at the distributions of values of something that can be measured you get a bell cure, like Brilliant drew here. The value most likely to turn up — the mode, mathematicians say — is also the arithmetic mean. “The average”, is what everybody except mathematicians say. And even they say that most of the time. But almost nobody is at the average.

Looking at a drawing, Brilliant’s included, explains why. The exact average is a tiny slice of all the data, the “population”. Look at the area in Brilliant’s drawing underneath the curve that’s just the blocks underneath the upside-down fellow. Most of the area underneath the curve is away from that.

There’s a lot of results that are close to but not exactly at the arithmetic mean. Most of the results are going to be close to the arithmetic mean. Look at how many area there is under the curve and within four vertical lines of the upside-down fellow. That’s nearly everything. So we have this apparent contradiction: the most likely result is the average. But almost nothing is average. And yet almost everything is nearly average. This is why statisticians have their own departments, or get to make the mathematics department brand itself the Department of Mathematics and Statistics.

## Reading the Comics, April 15, 2016: Remarkably, No Income Tax Comics Edition

I’m as startled as you are. While a couple comic strips mentioned United States Income Tax Day, they didn’t do so in a way that seemed on-point enough for this Reading The Comics post. Of course, United States Income Tax Day happens to be the 18th this year. I haven’t seen Sunday’s comics yet.

David L Hoyt and Jeff Knurek’s Jumble for the 11th of April one again uses arithmetic puns for its business. Also, if some science fiction writer doesn’t take hold of “Gribth” as a name for something they’re missing a fine syllable. “Tahew” is no slouch in the made-up word leagues either.

Ryan North’s Dinosaur Comics for the 12th of April obviously originally ran sometime in mid-March. I have similarly ambiguous feelings about the value of Pi Day. I suppose it’s nice for people to think of “fun” and “mathematics” close together. Utahraptor’s distinction between “Pi Day” of March 14 and “Approximate Pi Day” of the 22nd of July s a curious one, though. It’s not as though 3.14 is any more exactly π than 22/7 is. I suppose you can argue that at some moment on 3/14 between 1:59:26 and 1:59:27 there’s some moment, 1:59:26.5358979 et cetera going on forever. But that assumes that time is a continuous thing, and it’s not like you’ll ever know what that moment is. By the time you might recognize it, it’s passed. They are all Approximate Pi Days; we just have to decide what the approximation is.

Bill Schorr’s The Grizzwells for the 12th is a silly-homework problem question. I know the point is to joke about how Fauna misunderstands a word. But if we pretend the assignment is for real, what might its point be? To show that students know the parts of a right triangle? I guess that’s all right, but it doesn’t seem like much of an assignment. I don’t blame her for getting snarky in the face of that.

Rick Kirkman and Jerry Scott’s Baby Blues for the 13th is a gag about picking random numbers for arithmetic homework. The approach is doomed, surely, although it’s probably not completely doomed. I’m not sure Hammie’s age, but if his homework is about adding and subtracting numbers he probably mostly gets problems that give results between zero and twenty, and almost always less than a hundred. He might hit some by luck.

I’ve mentioned some how people are awful at picking “random” numbers in their heads. Zoe shows off one of the ways people are bad at it. People asked to name numbers “randomly” pick odd numbers more than even numbers. Somehow they just feel random. I doubt Kirkman and Scott were thinking of that; among other things, five numbers is a very small sample. Four odds out of five isn’t peculiar, not yet. They were probably just trying to pick numbers that sounded funny while fitting the space available. I’m a bit surprised 37 didn’t make the list.

Mark Anderson’s Andertoons for the 13th is Mark Anderson’s Andertoons entry for this essay. I like the teacher’s answer, though.

Patrick Roberts’s Todd the Dinosaur for the 14th just uses arithmetic as the most economic way to fit several problems on-screen at once. They’ve got a compactness that sentence-diagramming just can’t match.

Greg Cravens’s The Buckets for the 15th amuses me with its use of coin-tossing as a way of making choices. I’m also amused the coin might be wrong only about half the time.

John Deering’s Strange Brew for the 15th is a visual puzzle. It’s intending to make use of a board full of mathematical symbols to represent deep thought. But the symbols aren’t quite mathematics. They look much more like LaTeX, a typesetting code used to express mathematics in print. Some of the symbols are obscured, so I can’t say exactly what’s meant. But it should be something like this:

$F = \{F_{x} \in F_{c}: (is ... (1) ) \cap (minPixels < \|s\| < maxPixels ) \\ \partial{P} \\ (is_{connected}| > |s| - \epsilon) \}$

At the risk of disappointing, this appears to me gibberish. The appearance of words like ‘minPixels’ and ‘maxPixels’ suggest a bit of computer code. So does having a subscript that’s the full word “connected”. I wonder where Deering drew this example from.

## Reading the Comics, March 14, 2016: Pi Day Comics Event

Comic Strip Master Command had the regular pace of mathematically-themed comic strips the last few days. But it remembered what the 14th would be. You’ll see that when we get there.

Ray Billingsley’s Curtis for the 11th of March is a student-resists-the-word-problem joke. But it’s a more interesting word problem than usual. It’s your classic problem of two trains meeting, but rather than ask when they’ll meet it asks where. It’s just an extra little step once the time of meeting is made, but that’s all right by me. Anything to freshen the scenario up.

Tony Carrillo’s F Minus for the 11th was apparently our Venn Diagram joke for the week. I’m amused.

Mason Mastroianni, Mick Mastroianni, and Perri Hart’s B.C. for the 12th of March name-drops statisticians. Statisticians are almost expected to produce interesting pictures of their results. It is the field that gave us bar charts, pie charts, scatter plots, and many more. Statistics is, in part, about understanding a complicated set of data with a few numbers. It’s also about turning those numbers into recognizable pictures, all in the hope of finding meaning in a confusing world (ours).

Brian Anderson’s Dog Eat Doug for the 13th of March uses walls full of mathematical scrawl as signifier for “stuff thought deeply about’. I don’t recognize any of the symbols specifically, although some of them look plausibly like calculus. I would not be surprised if Anderson had copied equations from a book on string theory. I’d do it to tell this joke.

And then came the 14th of March. That gave us a bounty of Pi Day comics. Among them:

John Hambrock’s The Brilliant Mind of Edison Lee trusts that the name of the day is wordplay enough.

Scott Hilburn’s The Argyle Sweater is also a wordplay joke, although it’s a bit more advanced.

Tim Rickard’s Brewster Rockit fuses the pun with one of its running, or at least rolling, gags.

Bill Whitehead’s Free Range makes an urban legend out of the obsessive calculation of digits of π.

And Missy Meyer’s informational panel cartoon Holiday Doodles mentions that besides “National” Pi Day it was also “National” Potato Chip Day, “National” Children’s Craft Day, and “International” Ask A Question Day. My question: for the first three days, which nation?

Edited To Add: And I forgot to mention, after noting to myself that I ought to mention it. The Price Is Right (the United States edition) hopped onto the Pi Day fuss. It used the day as a thematic link for its Showcase prize packages, noting how you could work out π from the circumference of your new bicycles, or how π was a letter from your vacation destination of Greece, and if you think there weren’t brand-new cars in both Showcases you don’t know the game show well. Did anyone learn anything mathematical from this? I am skeptical. Do people come away thinking mathematics is more fun after this? … Conceivably. At least it was a day fairly free of people declaring they Hate Math and Can Never Do It.

## Terrible And Less-Terrible Things with Pi

We are coming around “Pi Day”, the 14th of March, again. I don’t figure to have anything thematically appropriate for the day. I figure to continue the Leap Day 2016 Mathematics A To Z, and I don’t tend to do a whole two posts in a single day. Two just seems like so many, doesn’t it?

But I would like to point people who’re interested in some π-related stuff to what I posted last year. Those posts were:

• Calculating Pi Terribly, in which I show a way to work out the value of π that’s fun and would take forever. I mean, yes, properly speaking they all take forever, but this takes forever just to get a couple of digits right. It might be fun to play with but don’t use this to get your digits of π. Really.
• Calculating Pi Less Terribly, in which I show a way to do better. This doesn’t lend itself to any fun side projects. It’s just calculations. But it gets you accurate digits a lot faster.

## Reading the Comics, March 9, 2016: Mathematics Recreation Edition

I haven’t been skipping the comics, even with the effort of keeping up on the Leap Day 2016 A To Z Glossary. I just try to keep to the pace which Comic Strip Master Command sets.

The kids-information feature Short Cuts, by Jeff Harris, got ahead of “Pi Day” last Sunday. I imagine the feature gets run mid-week in some features, so that it’s better to run a full week before March 14th. But here’s a bundle of trivia, some jokes, some activities, that sort of thing. I am curious about one of Harris’s trivias, that Pi “plays an important role in some of the equations used in Einstein’s famous general theory of relativity”. That’s true, but it’s not as if general relativity is a rare appearance for pi in physics. Maybe Harris chose it on aesthetic grounds. General relativity has a familiar name and exotic concepts. And it allowed him to put in an equation that’s mysterious yet attractive-looking.

Samson’s Dark Side Of The Horse for the 7th of March made me wonder how many sudoku puzzles there are. The answer is — well, you have to start thinking carefully about what you mean by “how many”. For example: start with one puzzle. Swap out every appearance of a 1 with a 2, and a 2 with a 1. Is this new one actually a different puzzle? You can make a case for yes or for no. And that’s before we get into the question of how many clues to give to solve the puzzle. If I’m not misreading Wikipedia’s “Mathematics of Sudoku” page, the number of different nine-by-nine combinations of digits that can be legitimate sudoku puzzle solutions is 6,670,903,752,021,072,936,960. This was worked out in 2005 by Bertram Felgenhauer and Frazer Jarvis. They worked it out partly by logic, partly by brute force. Brute force is trying all the possibilities to see what works. It’s a method that rewards endurance. We like that we can turn it over to computers now. Or cartoon horses, whichever. They’re good at endurance.

Jef Mallett’s Frazz started a sequence about problem-writing on the 7th of March. Caulfield’s setup, complaining about trains and apple bushels, suggests he was annoyed by mathematics problems. I understand. Much of real mathematics starts with curiosity about something (how many sudoku puzzles are there?). Then it’s working out what computation might answer that question. Then it’s doing that calculation. And then it’s verifying that the calculation is right. Mathematics educators have to teach ways to do a calculation, and test that. And to teach how to know what calculation to do, and test that. That’s challenging enough. Add to that working out something to be curious about and you understand the appeal of stock setups. Maybe mathematics should include some courses in creative writing and short-short fiction. (Verification is, in my experience, the part nobody cares about. This is a shame. The hardest part of doing numerical mathematics is making sure your computation makes any sense.)

Richard Thompson’s Richard’s Poor Almanac rerun the 7th of March features the Non-Euclidean Creeper. It’s a plant perhaps related to the Cubist Fir Christmas tree and to the Otterloops’ troublesome non-Euclidean tree. Non-Euclidean geometry will probably always sound more intimidating and exotic. Euclidean geometry describes the way objects on the human scale behave. Shapes that fit on the table, or in your garden, follow Euclidean rules. But non-Euclidean isn’t magic; it’s the way that shapes on the surface of a globe work, for example. And the idea of drawing a thing like a square on the surface of the Earth isn’t so bizarre.

Paul Trap’s Thatababy for the 7th makes sport of geometry.

My love and I were talking the other day about Jim Toomey’s Sherman’s Lagoon. It’s a bit odd as comic strips go. It’s been around forever, for one, but nobody talks about it. It’s stayed reliably funny. Comic strips that’ve been around forever tend to … you know … not be. The strip’s done as a work-and-home strip except the cast is all sea life. And the thing is, Toomey keeps paying attention to new discoveries in sea life, and other animal research. And this is a fantastic era for discoveries in sea life, aside from how humans have now eaten all of it and we don’t have any left. I am not joking when I say the comic strip is an effortless way to keep up with new discoveries about the oceans.

I missed it when in December the discovery was announced to the world. But the setup, about the common name being given by a group of kids, is apparently quite correct. So we should expect from Toomey. (The scientific name is Etmopterus benchleyi. The last name refers to Peter Benchley, repentant Jaws novelist.) LiveScience.com’s article says lead author Dr Vicky Vásquez had to “scale them back” from their starting point, the “super ninja”. This differs from Hawthorne’s claim that the kids started from the “math stinks” shark, but it’s still a delight anyway.

## Reading the Comics, October 17, 2015: Rerun Edition

I hate to make it sound like I’m running out of things to say about mathematical comics. But the most recent bunch of strips have been reruns, as with Bill Amend’s FoxTrot or Tom Toles’s Randolph Itch, 2 am. And there’s some figurative reruns too, as a couple of things I’ve talked about before come around again. Also I’m not sure but I think I might have used this Edition Title before. It feels like one I might have. I hope you’ll enjoy anyway, please.

Bill Amend’s FoxTrot Classics for the 15th of October, originally run in 2004, is about binary numerals. It’s built on the fact the numeral ‘100’ represents a rather smaller number in base-two arithmetic than it does in base-ten. This is the sort of thing that’s funny to a mathematically-inclined nerd, such as Jason here. It’s the numerical equivalent of a pun, playing on how if you pretend something is in a different context, it would have a different meaning.

Dave Blazek’s Loose Parts for the 15th of October puts a shape other than a triangle into the orchestra pit. I’m amused, and it puts me in mind of the classic question, “Can One Hear The Shape Of A Drum?” The answer is tricky.

Bob Scott’s Molly and the Bear for the 15th of October is a Pi Day joke. I don’t believe it’s a rerun, but the engagingly-drawn strip is in reruns terribly often.

Tom Toles’s Randolph Itch, 2 am for the 15th of October is a rerun, not just from 1999 but from earlier this year. I don’t know if the strip is being run out of order or if the strip ran a shorter time than I thought. Anyway, it’s still a funny drawing and “r” doesn’t figure into it at all.

Rick Detorie’s One Big Happy for the 16th of October shows Ruthie teaching her stuffed dolls about the number 1. Ruthie is a bit confused about the difference between the number one and the numeral, the way we represent the number. That’s common enough.

She does kind of have a point, though. The number one gets represented as a vertical stroke in the Arabic numerals we commonly use; also in Roman numerals used in making dates harder to read; also in Ancient Egyptian numerals; also in Chinese numerals. One almost suspects everyone is copying each other, or just started off with a tally mark and kept with it. Things get more complicated around ‘three’ or ‘four’. But it isn’t really universal, of course. The Mayans used a single dot, which is admittedly pretty close as a scheme. The Babylonians used a vertical wedge, a little triangle atop a stem that was presumably easy to carve with the tools available.

Ruben Bolling’s Super-Fun-Pak Comix for the 16th of October reprings a Chaos Butterfly installment. And the reminder that a system can be deterministic yet unpredictable sets me up for …

The rerun of Tom Toles’s Randolph Itch, 2 am that appeared on the 17th. The page of horoscopes saying “what happens to you today will be random, based on laws of probability” is funny, although, “random”? There is, it appears, randomness deeply encoded in the universe. There seems to be no way that atoms and molecules could work if they could not be random. But randomness follows laws. Those laws are so fundamental, and imply averages so relentlessly, that they create a human-scale world which might as well be deterministic. (I am deliberately bundling up the question of whether beings have free will and putting it off to the corner, in a little box, where I will not bother it.) In principle, we should be able to predict the day; we just need enough information, and time to compute.

Of course in practice we can’t, and can’t even come close. We may be able to predict the broad strokes of the day, but it is filled with the unpredictable. We call that random, but that is really a confession of ignorance. It’s much the way we might say there is a “probability” of one in seven that you were born on a Tuesday. There’s no such thing. The probability is either 1, because you were born on a Tuesday, or 0, because you were not. What day any given date in the Julian or Gregorian calendar occurred is a determined thing. What we mean by “a probability of one in seven” is that we are ignorant of your birthday, or have not done the work of finding out what day of the week that was. Thus the day of the week appears random.

John Graziano’s Ripley’s Believe It or Not for the 17th of October claims that Les Stewart wrote out “every number from one to one million in words’, using seven typewriters, in a project that took sixteen years and seven months. Sixteen years and seven months is something close to half a billion seconds. So if we take this, he was averaging about fifty seconds to write out each number. This sounds unimpressive, but after all, he had to take some time to sleep and probably had other projects to work on as well. Perhaps he was also working on putting the numbers in alphabetical order.

## Reading the Comics, October 10, 2015: Wordplay Edition

Some of the past several days’ mathematically-themed comic strips have bits of wordplay in them. That’ll do for the theme. We get some familiar topics along the way.

Rick Detorie’s One Big Happy for the 6th of October is one of the wordplay jokes you can do about probability. (This is the strip that ran in newspapers this year. One Big Happy strips on Gocomics.com are reruns from several years back.)

Niklas Eriksson’s Carpe Diem for the 6th of October is a badly-timed Pi Day strip.

Tom Thaves’s Frank and Ernest for the 8th of October is a kids-resisting-algebra problem. The kid asks why ‘x’ has to be equal to something, why it can’t just be ‘x’. He’s wiser than his teacher has taught. We use ‘x’ as the name for a number whose exact identity we don’t know right away. Often, especially in introductory algebra, we hope to work out what number it is. That’s the sort of problem that makes us find x, or solve for x. But we don’t always care what x is. Sometimes we just want to say that it’s an example of a number with some interesting properties. We often use it this way when we try drawing the plot of a function. The plot shows all the coordinate sets that make some equation true, and we need x to organize our thoughts about that, but we never really care what x is.

Or we might use x as a ‘dummy variable’, the mathematical equivalent of falsework. We use the variable to get some work done, but never see it once we’re finished, and don’t ever care what it was. If we take the definite integral of a function of x over x, for example, the one thing our answer should not have is an ‘x’ in it. (Well, if we’re integrating some nasty function that can’t be evaluated except in terms of another integral maybe an ‘x’ will appear. But that’s a pathological case.)

Alternatively, x might be a parameter, something which has to be a fixed number for the sake of doing other work, but whose value we don’t really care about. This would be an eccentric choice — usually parameters are from earlier in the alphabet, rarely later than ‘l’ and almost never past ‘t’ — but sometimes that’s the best alternative.

In Jef Mallett’s Frazz for the 8th of October, Caulfield answers his teacher’s demand to “show his work” by presenting a slide rule. It’s a cute joke although I’m not on Caulfield’s side here. If all anyone cared about was whether the calculation was right we’d need no mathematics. We have computers. What is worth teaching is “how do you know what to compute”, with a sideline of “can you do the computations correctly”. It’s important to know what you mean to do. It’s also important to know how to plausibly find an answer if you don’t know exactly what to do. None of that is shown by the answer alone.

Jim Benton’s Jim Benton Cartoons for the 8th of October is some more mathematics wordplay. I’m amused by its logic.

Samson’s Dark Side of the Horse for the 9th of October is the first anthropomorphized-numerals joke we’ve had in a while.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 9th of October is our Venn Diagram joke for this installment. And it’s not quite a proper Venn diagram, but it’s hard to draw a proper Venn diagram for four propositions. Wikipedia’s entry offers a couple of examples of four-set Venn diagrams. The one made of ellipses is not too bad, although it also evokes “logo for some maybe European cable TV channel” to my eye.

Disney’s Donald Duck for the 10th of October, a rerun from goodness knows when, depicts accurately the most terrifying moment a mathematician endures. I am delighted to see that the equations written out are correct and even consistent from one panel to the next. And yes, real mathematicians will sometimes write down what seem like altogether too-obvious propositions. That’s a good way of making sure you aren’t tripping over the easy stuff on the way to the bigger conclusions. I think it’s a bit implausible that the entire board would be this level of stuff — by the time you have your PhD, at least in mathematics or physics, you don’t need help remembering what the cosine of 120 degrees is — but it’s all valid stuff. Well, I could probably use the help remembering the tangent angle-addition formula, if I ever needed to work out the tangent of the sum of two angles.

## Reading the Comics, September 14, 2015: Back To School Edition, Part II

Today’s mathematical-comics blog should get us up to the present. Or at least to Monday. Yes, I’m aware of which paradox of Zeno this evokes. Be nice.

Scott Adams’s Dilbert Classics for the 11th of September is a rerun from, it looks like, the 18th of July, 1992. Anyway, Dilbert has acquired a supercomputer and figures to calculate π to a lot of decimal places, to finally do something about the areas of circles. The calculation of the digits of pi is done, often on home-brewed supercomputers, to lots of digits. But the calculation of areas, or volumes, or circumferences or whatever, isn’t why. We just don’t need that many digits; forty digits of pi would be plenty for almost any calculation involving measuring things in the real world.

It’s actually a bit mysterious why the digits of pi should be worth bothering with. It’s not yet known if pi is a “normal” number, and that’s of some interest. In a normal number every finite sequence of digits appears, and as often as every other sequence of digits just as long. That is, if you break the digits of pi up into four-digit blocks, you should see “1701” and “2038” and “2468” and “9999” and all, each appearing roughly once per thousand blocks. It’s almost certain that pi is normal, because just about every number is. But it’s not proven. And if there were numerical evidence that pi wasn’t normal that would be mathematically interesting, though I wouldn’t blame everybody not in number theory for saying “huh” and “so what?” before moving on. As it is, calculating digits of pi is a good, challenging task that can be used to prove coding and computing abilities, and it might turn up something interesting. It may as well be that as anything.

Steve Breen and Mike Thompson’s Grand Avenue for the 11th of September is a “motivate the word problem” joke. So is Bill Amend’s FoxTrot Classics for the 14th (a rerun from the same day in 2004). I like Amend’s version better, partly because it gives more realistic problems. I also like that it mixes in a bit of French class. It’s not always mathematics that needs motivation.

J C Duffy’s Lug Nuts for the 11th of September is another pun strip badly timed for Pi Day.

Ruben Bolling’s Tom The Dancing Bug gave us a Super Fun-Pak Comics installment on the 11th. And that included a Chaos Butterfly installment pitting deterministic chaos against Schrödinger’s Cat. The Cat represents one (of many) interpretations of quantum mechanics, the “superposition” interpretation. It’s difficult to explain the idea philosophically, to say what is really going on. The mathematics is straightforward, though. In the most common model of quantum mechanics we describe what is going on by a probability distribution, a function that describes how likely each possible outcome is. Quantum mechanics describes how that distribution changes in time. In the superpositioning we have two, or more, probability distributions that describe very different states, but (in a way) averaged together. The changes of this combined distribution then become our idea of how the system changes in time. It’s just hard to say what it could mean when the superposition implies very different things, like a cat being both wet and dry, being equally true at once.

Julie Larson’s Dinette Set for the 12th of September is about double negatives. It’s also about the doomed attempt to bring logic to the constructions of English. At least in English a double negative — “not unwanted”, say — generally parses to a positive, even if the connotation is that the thing is only a bit wanted. This corresponds to what logicians would say. A logican might use “C” to stand in for some statement that can only be true or false. Then, saying “not not C” — an “is true” gets implicitly added to the end of that — is equivalent to saying “C [is true]”. My love, the philosopher, who knows much more Spanish than I do has pointed out that in Spanish the “not not” construction can intensify the strength of the negation, rather than annulling it. This causes us to wonder if Spanish-speaking logic students have a harder time understanding the “not not C” construction. I don’t know and would welcome insight. (Also I hope I have it right that a “not not” is an intensifier, rather than a softener. But I suppose it doesn’t matter, as long as the Spanish equivalent of “not not wanted” still connotes “unwanted”.)

Dan Collins’s Looks Good On Paper for the 12th of September is a simple early-autumn panorama kind of strip. Mathematics — particularly, geometry — gets used as the type case for elementary school. I suppose as long as diagramming sentences is out of fashion there’s no better easy-to-draw choice.

David L Hoyt and Jeff Knurek’s Jumble for the 14th of September is an abacus joke. For folks who want to do the Jumble themselves, a hint: the second word is not “Dummy” however appealing an unscramble that looks.

Stephen Beals’s Adult Children for the 14th builds on the idea of what if the universe were made wrong. And that’s expressed as a mathematics error in the building of the universe. The idea of mathematics as a transcendent and even god-touching thing is an old one. I imagine this reflects how a logically deduced fact has some kind of universal truth, that a sound argument is sound regardless of who makes it, or considers it. It’s a heady idea. Mathematics also allows us to say some very specific, and remarkable, things about the infinite. This is another god-touching notion. But we don’t have sound reason to think that universe-making must be mathematical. Mathematics can describe many aspects of the universe eerily well, yes. But is it necessary that a universe be mathematically consistent? The question seems to defy any kind of empirical answer; we must listen to philosophers, who can at least give us a reasoned answer.

Tom Thaves’s Frank and Ernest for the 14th of September depicts cave-Frank and cave-Ernest at the dawn of numbers. It suggests the symbol 1 being a representation of a stick, and 0 as a stone. The 1 as a stick seems at least imaginable; counting off things by representing them as sticks or as stroke marks feels natural. Of course I say that coming from a long heritage of doing just that. 0, as I understand it, seems to derive from making with a dot a place where zero of whatever was to be studied should appear; the dot grew into a loop probably to make it harder to miss.

## Reading the Comics, August 29, 2015: Unthemed Edition

I can’t think of any particular thematic link through the past week’s mathematical comic strips. This happens sometimes. I’ll make do. They’re all Gocomics.com strips this time around, too, so I haven’t included the strips. The URLs ought to be reasonably stable.

J C Duffy’s Lug Nuts (August 23) is a cute illustration of the first, second, third, and fourth dimensions. The wall-of-text might be a bit off-putting, especially the last panel. It’s worth the reading. Indeed, you almost don’t need the cartoon if you read the text.

Tom Toles’s Randolph Itch, 2 am (August 24) is an explanation of pie charts. This might be the best stilly joke of the week. I may just be an easy touch for a pie-in-the-face.

Charlie Podrebarac’s Cow Town (August 26) is about the first day of mathematics camp. It’s also every graduate students’ thesis defense anxiety dream. The zero with a slash through it popping out of Jim Smith’s mouth is known as the null sign. That comes to us from set theory, where it describes “a set that has no elements”. Null sets have many interesting properties considering they haven’t got any things. And that’s important for set theory. The symbol was introduced to mathematics in 1939 by Nicholas Bourbaki, the renowned mathematician who never existed. He was important to the course of 20th century mathematics.

Eric the Circle (August 26), this one by ‘Arys’, is a Venn diagram joke. It makes me realize the Eric the Circle project does less with Venn diagrams than I expected.

John Graziano’s Ripley’s Believe It Or Not (August 26) talks of a Akira Haraguchi. If we believe this, then, in 2006 he recited 111,700 digits of pi from memory. It’s an impressive stunt and one that makes me wonder who did the checking that he got them all right. The fact-checkers never get their names in Graziano’s Ripley’s.

Mark Parisi’s Off The Mark (August 27, rerun from 1987) mentions Monty Hall. This is worth mentioning in these parts mostly as a matter of courtesy. The Monty Hall Problem is a fine and imagination-catching probability question. It represents a scenario that never happened on the game show Let’s Make A Deal, though.

Jeff Stahler’s Moderately Confused (August 28) is a word problem joke. I do wonder if the presence of battery percentage indicators on electronic devices has helped people get a better feeling for percentages. I suppose only vaguely. The devices can be too strangely nonlinear to relate percentages of charge to anything like device lifespan. I’m thinking here of my cell phone, which will sit in my messenger bag for three weeks dropping slowly from 100% to 50%, and then die for want of electrons after thirty minutes of talking with my father. I imagine you have similar experiences, not necessarily with my father.

Thom Bluemel’s Birdbrains (August 29) is a caveman-mathematics joke. This one’s based on calendars, which have always been mathematical puzzles.

## Reading the Comics, June 25, 2015: Not Making A Habit Of This Edition

I admit I did this recently, and am doing it again. But I don’t mean to make it a habit. I ran across a few comic strips that I can’t, even with a stretch, call mathematically-themed, but I liked them too much to ignore them either. So they’re at the end of this post. I really don’t intend to make this a regular thing in Reading the Comics posts.

Justin Boyd’s engagingly silly Invisible Bread (June 22) names the tuning “two steps below A”. He dubs this “negative C#”. This is probably an even funnier joke if you know music theory. The repetition of the notes in a musical scale could be used as an example of cyclic or modular arithmetic. Really, that the note above G is A of the next higher octave, and the note below A is G of the next lower octave, probably explains the idea already.

If we felt like, we could match the notes of a scale to the counting numbers. Match A to 0, B to 1, C to 2 and so on. Work out sharps and flats as you like. Then we could think of transposing a note from one key to another as adding or subtracting numbers. (Warning: do not try to pass your music theory class using this information! Transposition of keys is a much more subtle process than I am describing.) If the number gets above some maximum, it wraps back around to 0; if the number would go below zero, it wraps back around to that maximum. Relabeling the things in a group might make them easier or harder to understand. But it doesn’t change the way the things relate to one another. And that’s why we might call something F or negative C#, as we like and as we hope to be understood.

Hilary Price’s Rhymes With Orange (June 23) reminds us how important it is to pick the correct piece of chalk. The mathematical symbols on the board don’t mean anything. A couple of the odder bits of notation might be meant as shorthand. Often in the rush of working out a problem some of the details will get written as borderline nonsense. The mathematician is probably more interested in getting the insight down. She’ll leave the details for later reflection.

Jason Poland’s Robbie and Bobby (June 23) uses “calculating obscure digits of pi” as computer fun. Calculating digits of pi is hard, at least in decimals, which is all anyone cares about. If you wish to know the 5,673,299,925th decimal digit of pi, you need to work out all 5,673,299,924 digits that go before it. There are formulas to work out a binary (or hexadecimal) digit of pi without working out all the digits that go before. This saves quite some time if you need to explore the nether-realms of pi’s digits.

The comic strip also uses Stephen Hawking as the icon for most-incredibly-smart-person. It’s the role that Albert Einstein used to have, and still shares. I am curious whether Hawking is going to permanently displace Einstein as the go-to reference for incredible brilliance. His pop culture celebrity might be a transient thing. I suspect it’s going to last, though. Hawking’s life has a tortured-genius edge to it that gives it Romantic appeal, likely to stay popular.

Paul Trap’s Thatababy (June 23) presents confusing brand-new letters and numbers. Letters are obviously human inventions though. They’ve been added to and removed from alphabets for thousands of years. It’s only a few centuries since “i” and “j” became (in English) understood as separate letters. They had been seen as different ways of writing the same letter, or the vowel and consonant forms of the same letter. If enough people found a proposed letter useful it would work its way into the alphabet. Occasionally the ampersand & has come near being a letter. (The ampersand has a fascinating history. Honestly.) And conversely, if we collectively found cause to toss one aside we could remove it from the alphabet. English hasn’t lost any letters since yogh (the Old English letter that looks like a 3 written half a line off) was dropped in favor of “gh”, about five centuries ago, but there’s no reason that it couldn’t shed another.

Numbers are less obviously human inventions. But the numbers we use are, or at least work like they are. Arabic numerals are barely eight centuries old in Western European use. Their introduction was controversial. People feared shopkeepers and moneylenders could easily cheat people unfamiliar with these crazy new symbols. Decimals, instead of fractions, were similarly suspect. Negative numbers took centuries to understand and to accept as numbers. Irrational numbers too. Imaginary numbers also. Indeed, look at the connotations of those names: negative numbers. Irrational numbers. Imaginary numbers. We can add complex numbers to that roster. Each name at least sounds suspicious of the innovation.

There are more kinds of numbers. In the 19th century William Rowan Hamilton developed quaternions. These are 4-tuples of numbers that work kind of like complex numbers. They’re strange creatures, admittedly, not very popular these days. Their greatest strength is in representing rotations in three-dimensional space well. There are also octonions, 8-tuples of numbers. They’re more exotic than quaternions and have fewer good uses. We might find more, in time.

Rina Piccolo’s entry in Six Chix this week (June 24) draws a house with extra dimensions. An extra dimension is a great way to add volume, or hypervolume, to a place. A cube that’s 20 feet on a side has a volume of 203 or 8,000 cubic feet, after all. A four-dimensional hypercube 20 feet on each side has a hypervolume of 160,000 hybercubic feet. This seems like it should be enough for people who don’t collect books.

Morrie Turner’s Wee Pals (June 24, rerun) is just a bit of wordplay. It’s built on the idea kids might not understand the difference between the words “ratio” and “racial”.

Tom Toles’s Randolph Itch, 2 am (June 25, rerun) inspires me to wonder if anybody’s ever sold novelty 4-D glasses. Probably they have, sometime.

Now for the comics that I just can’t really make mathematics but that I like anyway:

Phil Dunlap’s Ink Pen (June 23, rerun) is aimed at the folks still lingering in grad school. Please be advised that most doctoral theses do not, in fact, end in supervillainy.

Darby Conley’s Get Fuzzy (June 25, rerun) tickles me. But Albert Einstein did after all say many things in his life, and not everything was as punchy as that line about God and dice.

## Error

This is one of my A to Z words that everyone knows. An error is some mistake, evidence of our human failings, to be minimized at all costs. That’s … well, it’s an attitude that doesn’t let you use error as a tool.

An error is the difference between what we would like to know and what we do know. Usually, what we would like to know is something hard to work out. Sometimes it requires complicated work. Sometimes it requires an infinite amount of work to get exactly right. Who has the time for that?

This is how we use errors. We look for methods that approximate the thing we want, and that estimate how much of an error that method makes. Usually, the method involves doing some basic step some large number of times. And usually, if we did the step more times, the estimate of the error we make will be smaller. My essay “Calculating Pi Less Terribly” shows an example of this. If we add together more terms from that Leibniz formula we get a running total that’s closer to the actual value of π.