Reading the Comics, April 24, 2017: Reruns Edition


I went a little wild explaining the first of last week’s mathematically-themed comic strips. So let me split the week between the strips that I know to have been reruns and the ones I’m not so sure were.

Bill Amend’s FoxTrot for the 23rd — not a rerun; the strip is still new on Sundays — is a probability question. And a joke about story problems with relevance. Anyway, the question uses the binomial distribution. I know that because the question is about doing a bunch of things, homework questions, each of which can turn out one of two ways, right or wrong. It’s supposed to be equally likely to get the question right or wrong. It’s a little tedious but not hard to work out the chance of getting exactly six problems right, or exactly seven, or exactly eight, or so on. To work out the chance of getting six or more questions right — the problem given — there’s two ways to go about it.

One is the conceptually easy but tedious way. Work out the chance of getting exactly six questions right. Work out the chance of getting exactly seven questions right. Exactly eight questions. Exactly nine. All ten. Add these chances up. You’ll get to a number slightly below 0.377. That is, Mary Lou would have just under a 37.7 percent chance of passing. The answer’s right and it’s easy to understand how it’s right. The only drawback is it’s a lot of calculating to get there.

So here’s the conceptually harder but faster way. It works because the problem says Mary Lou is as likely to get a problem wrong as right. So she’s as likely to get exactly ten questions right as exactly ten wrong. And as likely to get at least nine questions right as at least nine wrong. To get at least eight questions right as at least eight wrong. You see where this is going: she’s as likely to get at least six right as to get at least six wrong.

There’s exactly three possibilities for a ten-question assignment like this. She can get four or fewer questions right (six or more wrong). She can get exactly five questions right. She can get six or more questions right. The chance of the first case and the chance of the last have to be the same.

So, take 1 — the chance that one of the three possibilities will happen — and subtract the chance she gets exactly five problems right, which is a touch over 24.6 percent. So there’s just under a 75.4 percent chance she does not get exactly five questions right. It’s equally likely to be four or fewer, or six or more. Just-under-75.4 divided by two is just under 37.7 percent, which is the chance she’ll pass as the problem’s given. It’s trickier to see why that’s right, but it’s a lot less calculating to do. That’s a common trade-off.

Ruben Bolling’s Super-Fun-Pax Comix rerun for the 23rd is an aptly titled installment of A Million Monkeys At A Million Typewriters. It reminds me that I don’t remember if I’d retired the monkeys-at-typewriters motif from Reading the Comics collections. If I haven’t I probably should, at least after making a proper essay explaining what the monkeys-at-typewriters thing is all about.

'This new math teacher keeps shakin' us down every morning, man ... what's she looking for, anyway?' 'Pocket calculators.'
Ted Shearer’s Quincy from the 28th of February, 1978. So, that FoxTrot problem I did? The conceptually-easy-but-tedious way is not too hard to do if you have a calculator. It’s a buch of typing but nothing more. If you don’t have a calculator, though, the desire not to do a whole bunch of calculating could drive you to the conceptually-harder-but-less-work answer. Is that a good thing? I suppose; insight is a good thing to bring. But the less-work answer only works because of a quirk in the problem, that Mary Lou is supposed to have a 50 percent chance of getting a question right. The low-insight-but-tedious problem will aways work. Why skip on having something to do the tedious part?

Ted Shearer’s Quincy from the 28th of February, 1978 reveals to me that pocket calculators were a thing much earlier than I realized. Well, I was too young to be allowed near stuff like that in 1978. I don’t think my parents got their first credit-card-sized, solar-powered calculator that kind of worked for another couple years after that. Kids, ask about them. They looked like good ideas, but you could use them for maybe five minutes before the things came apart. Your cell phone is so much better.

Bil Watterson’s Calvin and Hobbes rerun for the 24th can be classed as a resisting-the-word-problem joke. It’s so not about that, but who am I to slow you down from reading a Calvin and Hobbes story?

Garry Trudeau’s Doonesbury rerun for the 24th started a story about high school kids and their bad geography skills. I rate it as qualifying for inclusion here because it’s a mathematics teacher deciding to include more geography in his course. I was amused by the week’s jokes anyway. There’s no hint given what mathematics Gil teaches, but given the links between geometry, navigation, and geography there is surely something that could be relevant. It might not help with geographic points like which states are in New England and where they are, though.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 24th is built on a plot point from Carl Sagan’s science fiction novel Contact. In it, a particular “message” is found in the digits of π. (By “message” I mean a string of digits that are interesting to us. I’m not sure that you can properly call something a message if it hasn’t got any sender and if there’s not obviously some intended receiver.) In the book this is an astounding thing because the message can’t be; any reasonable explanation for how it should be there is impossible. But short “messages” are going to turn up in π also, as per the comic strips.

I assume the peer review would correct the cartoon mathematicians’ unfortunate spelling of understanding.

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, May 18, 2014: Pop Math of the 80s Edition


And now there’ve suddenly been enough mathematics-themed comics for a fresh collection of the things. If there’s any theme this time around it’s to mathematics I remember filtering into popular culture in the 80s: the Drake Equation (which I, at least, first saw in Carl Sagan’s Cosmos and found haunting), and the Rubik’s Cube, which pop mathematics writers in the early 80s latched onto with an eagerness matched only by how they liked polyominoes in the mid-70s, and the Mandelbrot Set, which I think of as a mid-to-late 80s thing because that’s when it started covering science-oriented magazine covers and the screens of IBM PS/2’s being used by the kids in the math and science magnet programs.

Incidentally, this time around I’ve tried to include the Between Friends that I talk about, because I’m not convinced the link to its Comics Kingdom home site will last indefinitely. Gocomics.com seems to keep links from expiring, even for non-subscribers, but I’m curious whether it would be better-liked if I included images of the strips I talk about? I’m fairly confident that this is fair use, as I talk about mathematical subjects inspired by the strips, but I don’t know whether people care much about saving a click before reading my attempts to say something, anything, about a kid given a word problem about airplanes that he answers in a flippant manner.

Wulff and Morgenthaler’s WuMo (May 15) features Professor Rubik, “five minutes after” inventing what he’s famous for. Ernö Rubik really is a Professor (of architecture, at the Budapest College of Applied Arts when he invented his famous cube), and was interested in the relationships of things in space and of objects moving in space. The Rubik’s Cube is of interest mathematically because it offers a great excuse to introduce group theory to the average person. Group theory is, among other things, a way of studying structures that look like arithmetic but that aren’t necessarily on numbers. Rotations work very much like the addition of numbers, at least, the modular addition (where if a result is less than zero, or greater than some upper bound, you add or subtract that upper bound until the result is back in range), and the Rubik’s Cube offers several interacting sets of things to rotate, so that the groups represented by it are fascinatingly complex.

Though the cube was invented in 1974 it didn’t become an overwhelming phenomenon until 1979, and then much of the early 80s was spent in people making jokes about how frustrating they found it and occasionally buying books that were supposed to tell you how to solve it, but you couldn’t after all. Then there was a Saturday morning cartoon about the cube which I watched because I had horrible, horrible, horrible taste in cartoons as a kid. Anyway, it turns out that if you played it perfectly you could solve any Rubik’s Cube in no more than twenty steps, although this wasn’t proven until 2010. I confess I usually just give up around step 35 and take the cube apart. Don’t watch the cartoon.

Eric the Circle (May 17), this entry by “Designroo”, features Eric in the midst of the Mandelbrot Set. The Mandelbrot Set, basis for two-thirds of all the posters on the walls in the mathematics department from 1986 through 2002, was discovered by Benoît Mandelbrot in one of those triumphs of numerical computing. It’s not hard to describe how to make it — it’s only a little more advanced than the pastime of hitting a square root or a square button on a calculator and watching numbers dwindle to zero or grow infinitely large — but the number of calculations that need to be done to see it mean it’d never have been discovered before there were computers to do the hard work, of calculation and of visualization.

Among the neat things about the Mandelbrot set are that it does have inlets that look like circles, and it has an infinite number of them: if you zoom in closely at any point on the boundary of the Mandelbrot set you’ll find a not-quite-perfect replica of the original set, with the big carotid shape and the budding circles on the edges, over and over, inexhaustibly.

Bill Amend’s FoxTrot (May 17, rerun) asks why there aren’t geometry books on tape. It’s not quite an absurd question: in principle, geometry is a matter of deductive logic, and is about the relationship between ideas we call “points” and “lines” and “angles” and the like. Pictures are nice to have, as appeals to intuition, but our intuition can be wrong, and pictures can lead us astray, as any optical illusion will prove. And yet it’s so very hard to do away with that intuition. We may not know a compelling reason why the things we draw on sheets of paper should correspond to the results of logical, deductive reasoning that ought to be true whether drawn or not and whether, for that matter, a universe existed or not, but seeing representations of the relationships of geometric objects seems to help nearly everyone understand them better than simply knowing the reasons they should have those relationships.

The notion of learning geometry without drawings takes one fairly close to the Bourbaki project, the famous/infamous early 20th century French mathematical collective that tried to work out the logical structure of all mathematics on a purely formal, reasoned basis without any appeals to diagrams or physical intuition at all. It was an ambitious, controversial, and fruitful program that got permanently tainted because following from it was the “New Math”, an attempt at mathematics educational reform of the 60s and 70s which crashed hard against the problem that parents will only support educational reform that doesn’t involve teaching a thing in ways different from how they learned it.

Sondra Bell-Lundy's _Between Friends_, 18 May 2014: mothers draw a Venn Diagram.
Sondra Bell-Lundy’s _Between Friends_, 18 May 2014: mothers draw a Venn Diagram.

Sandra Bell-Lundy’s Between Friends (May 18) brings up my old nemesis of Venn Diagrams, although it gets them correct.

T Lewis and Michael Fry’s Over The Hedge (May 18) showcases the Drake Equation, a wonderful bit of reasoning that tries to answer the question of “how many species capable of interstellar communication are there”, considering that we only have evidence for at most one. It’s a wonderful bit of word-problem-type reasoning: given what we do know, which amounts mostly to how many stars there are, how can we work out what we would like to know? Frank Drake, astronomer, and co-designer of the plaque on Pioneers 10 and 11, made some estimates of what factors are relevant in going from what we do know to what we would like to know, and how they might relate. When Drake first published the equation only the number of stars could be reasonably estimated; today we can also add a good estimate of how likely a star is to have planets, and a fair estimate of how likely a planet is to be livable. The other steps are harder to estimate. But the process Drake used, of evaluating what he would need to know in order to give an answer, is still strong: there may be things about the equation which are wrong — factors that interact in ways not previously considered, for example — but it divides a huge problem into a series of smaller ones that can, hopefully, be studied and understood in pieces and through this process be turned into knowledge.

And finally, Jeff Harris’s Shortcuts (May 18), a kid’s activity/information panel, spends a half-a-comics-page talking about numbers and numerals. It’s a pretty respectable short guide to numbers and their representations, including some of the more famous number-representation schemes.