Calculating Pi Less Terribly


Back on “Pi Day” I shared a terrible way of calculating the digits of π. It’s neat in principle, yes. Drop a needle randomly on a uniformly lined surface. Keep track of how often the needle crosses over a line. From this you can work out the numerical value of π. But it’s a terrible method. To be sure that π is about 3.14, rather than 3.12 or 3.38, you can expect to need to do over three and a third million needle-drops. So I described this as a terrible way to calculate π.

A friend on Twitter asked if it was worse than adding up 4 * (1 – 1/3 + 1/5 – 1/7 + … ). It’s a good question. The answer is yes, it’s far worse than that. But I want to talk about working π out that way.

Science teacher Mark Twain says if you divide the diameter of the moon into its circumference you get pi in the sky, and then laughs so hard at his own joke it causes chest pains.
Tom Batiuk’s Funky Winkerbean for the 17th of May, 2015. The worst part of this strip is Science Teacher Mark Twain will go back to the teachers’ lounge and complain that none of his students got it.

This isn’t part of the main post. But the comic strip happened to mention π on a day when I’m talking about π so who am I to resist coincidence?


Continue reading “Calculating Pi Less Terribly”

Reading the Comics, April 15, 2015: Tax Day Edition


Since it is mid-April, and most of the comic strips at Comics Kingdom and GoComics.com are based in the United States, Comic Strip Master Command ordered quite a few comics about taxes. Most of those are simple grumbling, but the subject naturally comes around to arithmetic and calculation and sometimes even logic. Thus, this is a Tax Day edition, though it’s bookended with Mutt and Jeff.

Bud Fisher’s Mutt And Jeff (April 11) — a rerun rom goodness only knows when, and almost certainly neither written nor drawn by Bud Fisher at that point — recounts a joke that has the form of a word problem in which a person’s age is deduced from information about the age. It’s an old form, but jokes about cutting the Gordion knot are probably always going to be reliable. I’m reminded there’s a story of Thomas Edison giving a new hire, mathematician, the problem of working out the volume of a light bulb. Edison got impatient with the mathematician treating it as a calculus problem — the volume of a rotationally symmetric object like a bulb is the sort of thing you can do by the end of Freshman Calculus — and instead filling a bulb with water, pouring the water into a graduated cylinder, and reading it off that.

Calculus under 50: vectors and stuff. Calculus over 50: diet and exercise problems.
Sandra Bell-Lundy’s Between Friends for the 12th of April, 2015. The link will likely expire around the 12th of May.

Sandra Bell-Lundy’s Between Friends (April 12) uses Calculus as the shorthand for “the hardest stuff you might have to deal with”. The symbols on the left-hand side are fair enough, although I’d think of them more as precalculus or linear algebra or physics, but they do parse well enough as long as I suppose that what sure looks like a couple of extraneous + signs are meant to refer to “t”. But “t” is a common enough variable in calculus problems, usually representing time, sometimes just representing “some parameter whose value we don’t really care about, but we don’t want it to be x”, and it looks an awful lot like a plus sign there too. On the right side, I have no idea what a root of forty minutes on a treadmill might be. It’s symbolic.

Continue reading “Reading the Comics, April 15, 2015: Tax Day Edition”

Reading the Comics, April 6, 2015: Little Infinite Edition


As I warned, there were a lot of mathematically-themed comic strips the last week, and here I can at least get us through the start of April. This doesn’t include the strips that ran today, the 7th of April by my calendar, because I have to get some serious-looking men to look at my car and I just know they’re going to disapprove of what my CV joint covers look like, even though I’ve done nothing to them. But I won’t be reading most of today’s comic strips until after that’s done, and so commenting on them later.

Mark Anderson’s Andertoons (April 3) makes its traditional appearance in my roundup, in this case with a business-type guy declaring infinity to be “the loophole of all loopholes!” I think that’s overstating things a fair bit, but strange and very counter-intuitive things do happen when you try to work out a problem in which infinities turn up. For example: in ordinary arithmetic, the order in which you add together a bunch of real numbers makes no difference. If you want to add together infinitely many real numbers, though, it is possible to have them add to different numbers depending on what order you add them in. Most unsettlingly, it’s possible to have infinitely many real numbers add up to literally any real number you like, depending on the order in which you add them. And then things get really weird.

Keith Tutt and Daniel Saunders’s Lard’s World Peace Tips (April 3) is the other strip in this roundup to at least name-drop infinity. I confess I don’t see how “being infinite” would help in bringing about world peace, but I suppose being finite hasn’t managed the trick just yet so we might want to think outside the box.

Continue reading “Reading the Comics, April 6, 2015: Little Infinite Edition”

My Mathematics Blog, As March 2015 Would Have It


And now for my monthly review of publication statistics. This is a good month to do it with, since it was a record month: I had 1,022 pages viewed around these parts, the first time (according to WordPress) that I’ve had more than a thousand in a month. In January I’d had 944, and in February a mere 859, which I was willing to blame on the shortness of that month. March’s is a clean record, though, more views per day than either of those months.

The total number of visitors was up, too, to 468. That’s compared to 438 in January and 407 in short February, although it happens it’s not a record; that’s still held by January 2013 and its 473 visitors. The number of views per visitor keeps holding about steady: from 2.16 in January to 2.11 in February to 2.18 in March. It appears that I’m getting a little better at finding people who like to read what I like to write, but haven’t caught that thrilling transition from linear to exponential growth.

The new WordPress statistics tell me I had a record 265 likes in March, up from January’s 196 and February’s 179. The number of comments rose from January’s 51 and February’s 56 to a full 93 for March. I take all this as supporting evidence that I’m better at reaching people lately. (Although I do wonder if it counts backlinks from one of my articles to another as a comment.)

The mathematics blog starts the month at 22,837 total views, and with 454 WordPress followers.

The most popular articles in March, though, were the set you might have guessed without actually reading things around here:

I admit I thought the “how interesting is a basketball tournament?” thing would be more popular, but it’s hampered by having started out in the middle of the month. I might want to start looking at the most popular articles of the past 30 days in the middle of the month too.

The countries sending me the greatest number of readers were the usual set: the United States at 658 in first place, and Canada in second at 66. The United Kingdom was a strong third at 57, and Austria in fourth place at 30.

Sending me a single reader each were Belgium, Ecuador, Israel, Japan, Lebanon, Mexico, Nepal, Norway, Portugal, Romania, Samoa, Saudi Arabia, Slovakia, Thailand, the United Arab Emirates, Uruguay, and Venezuela. The repeats from February were Japan, Mexico, Romania, and Venezuela. Japan is on a three-month streak, while Mexico has sent me a solitary reader four months in a row. India’s declined slightly in reading me, from 6 to 5. Ah well.

Among the interesting search terms were:

  • right trapezoid 5 (I loved this anime as a kid)
  • a short comic strip on reminding people on how to order decimals correctly (I hope they found what they were looking for)
  • are there other ways to draw a trapezoid (try with food dye on the back of your pet rabbit!)
  • motto of ideal gas (veni vidi v = nRT/P ?)
  • rectangular states (the majority of United States states are pretty rectangular, when you get down to it)
  • what is the definition of rerun (I don’t think this has come up before)
  • what are the chances of consecutive friday the 13th’s in a year (I make it out at 3/28, or a touch under 11 percent; anyone have another opinion?)

Well, with luck, I should have a fresh comic strips post soon and some more writing in the curious mix between information theory and college basketball.

Reading the Comics, March 26, 2015: Kind Of Hanging Around Edition


I’m sorry to have fallen silent the last few days; it’s been a bit busy and I’ve been working on follow-ups to a couple of threads. Fortunately Comic Strip Master Command is still around and working to make sure I don’t disappear altogether, and I have a selection of comic strips which at least include a Jumble world puzzle, which should be a fun little diversion.

Tony Rubino and Gary Markstein’s Daddy’s Home (March 23) asks what seems like a confused question to me, “if you believe in infinity, does that mean anything is possible?” As I say, I’m not sure I understand how belief in infinity comes into play, but that might just reflect my background: I’ve been thoroughly convinced that one can describe collections of things that have infinitely many elements — the counting numbers, rectangles, continuous functions — as well as that one can subdivide things — like segments of a number line — infinitely many times — as well as of quantities that are larger than any finite number and so must be infinitely large; so, what’s to not believe in? (I’m aware that there are philosophical and theological questions that get into things termed “potential” and “actual” infinities, but I don’t understand the questions those terms are meant to address.) The phrasing of “anything is possible” seems obviously flawed to me. But if we take it to mean instead “anything not logically inconsistent or physically prohibited is possible” then we seem to have a reasonable question, if that hasn’t just reduced to “anything not impossible is possible”. I guess ultimately I just wonder if the kid is actually trying to understand anything or if he’s just procrastinating.

Continue reading “Reading the Comics, March 26, 2015: Kind Of Hanging Around Edition”

Reading the Comics, March 15, 2015: Pi Day Edition


I had kind of expected the 14th of March — the Pi Day Of The Century — would produce a flurry of mathematics-themed comics. There were some, although they were fewer and less creatively diverse than I had expected. Anyway, between that, and the regular pace of comics, there’s plenty for me to write about. Recently featured, mostly on Gocomics.com, a little bit on Creators.com, have been:

Brian Anderson’s Dog Eat Doug (March 11) features a cat who claims to be “pondering several quantum equations” to prove something about a parallel universe. It’s an interesting thing to claim because, really, how can the results of an equation prove something about reality? We’re extremely used to the idea that equations can model reality, and that the results of equations predict real things, to the point that it’s easy to forget that there is a difference. A model’s predictions still need some kind of validation, reason to think that these predictions are meaningful and correct when done correctly, and it’s quite hard to think of a meaningful way to validate a predication about “another” universe.

Continue reading “Reading the Comics, March 15, 2015: Pi Day Edition”

Calculating Pi Terribly


I’m not really a fan of Pi Day. I’m not fond of the 3/14 format for writing dates to start with — it feels intolerably ambiguous to me for the first third of the month — and it requires reading the / as a . to make sense, when that just is not how the slash works. To use the / in any of its normal forms then Pi Day should be the 22nd of July, but that’s incompatible with the normal American date-writing conventions and leaves a day that’s nominally a promotion of the idea that “mathematics is cool” in the middle of summer vacation. This particular objection evaporates if you use . as the separator between month and day, but I don’t like that either, since it uses something indistinguishable from a decimal point as something which is not any kind of decimal point.

Also it encourages people to post a lot of pictures of pies, and make jokes about pies, and that’s really not a good pun. It plays on the coincidence of sounds without having any of the kind of ambiguity or contrast between or insight into concepts that normally make for the strongest puns, and it hasn’t even got the spontaneity of being something that just came up in conversation. We could use better jokes is my point.

But I don’t want to be relentlessly down about what’s essentially a bit of whimsy. (Although, also, dropping the ’20’ from 2015 so as to make this the Pi Day Of The Century? Tom Servo has a little song about that sort of thing.) So, here’s a neat and spectacularly inefficient way to generate the value of pi, that doesn’t superficially rely on anything to do with circles or diameters, and that’s probability-based. The wonderful randomness of the universe can give us a very specific and definite bit of information.

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Reading the Comics, January 17, 2015: Finding Your Place Edition


This week’s collection of mathematics-themed comic strips includes one of the best examples of using mathematics in real life, because it describes how to find your position if you’re lost in, in this case, an uncharted island. I’m only saddened that I couldn’t find a natural way to work in how to use an analog watch as a makeshift compass, so I’m shoehorning it in up here, as well as pointing out that if you don’t have an analog clock to use, you can still approximate it by drawing the hands of the clock on a sheet of paper and using that as a pretend watch, and there is something awesome about using a sheet of paper with the time drawn on it as a way to finding north.

Dave Whamond’s Reality Check (January 12) is a guru-on-the-mountain joke, explaining that the answers to life are in the back of the math book. It’s certainly convention for a mathematics book, at least up through about Intro Differential Equations, to include answers to the problems, or at least a selection of problems, in the back, and on reflection it’s a bit of an odd convention. You don’t see that in, say, a history book even where the questions can be reduced to picking out trivia from the main text. I suppose the math-answers convention reflects an idea that there’s a correct way to go about solving a problem, and therefore, you can check whether you picked the correct way and followed it correctly with no more answer than a printed “15/2” as guide. In this way, I suppose, a mathematics textbook can be self-teaching — at least, the eager student can do some of her own pass/fail grading — which was probably invaluable back in the days when finding a skilled mathematics teacher was so much harder than it is today.

Continue reading “Reading the Comics, January 17, 2015: Finding Your Place Edition”

Reading the Comics, December 30, 2014: Surely This Is It For The Year Edition?


Well, I thought it’d be unlikely to get too many more mathematics comics before the end of the year, but Comic Strip Master Command apparently sent out orders to clear out the backlog before the new calendar year starts. I think Dark Side of the Horse is my favorite of the strips, blending a good joke with appealing artwork, although The Buckets gives me the most to talk about.

Greg Cravens’s The Buckets (December 28) is about what might seem only loosely a mathematical topic: that the calendar is really a pretty screwy creation. And it is, as anyone who’s tried to program a computer to show dates has realized. The core problem, I suppose, is that the calendar tries to meet several goals simultaneously: it’s supposed to use our 24-hour days to keep track of the astronomical year, which is an approximation to the cycle of seasons of the year, and there’s not a whole number of days in a year. It’s also supposed to be used to track short-term events (weeks) and medium-term events (months and seasons). The number of days that best approximate the year, 365 and 366, aren’t numbers that lend themselves to many useful arrangements. The months try to divide that 365 or 366 reasonably uniformly, with historial artifacts that can be traced back to the Roman calendar was just an unspeakable mess; and, something rarely appreciated, the calendar also has to make sure that the date of Easter is something reasonable. And, of course, any reforming of the calendar has to be done with the agreement of a wide swath of the world simultaneously. Given all these constraints it’s probably remarkable that it’s only as messed up as it is.

To the best of my knowledge, January starts the New Year because Tarquin Priscus, King of Rome from 616 – 579 BC, found that convenient after he did some calendar-rejiggering (particularly, swapping the order of February and January), though I don’t know why he thought that particularly convenient. New Years have appeared all over the calendar year, though, with the start of January, the start of September, Christmas Day, and the 25th of March being popular options, and if you think it’s messed up to have a new year start midweek, think about having a new year start in the middle of late March. It all could be worse.

Continue reading “Reading the Comics, December 30, 2014: Surely This Is It For The Year Edition?”

Reading the Comics, December 14, 2014: Pictures Gone Again? Edition


I’ve got enough comics to do a mathematics-comics roundup post again, but none of them are the King Features or Creators or other miscellaneous sources that demand they be included here in pictures. I could wait a little over three hours and give the King Features Syndicate comics another chance to say anything on point, or I could shrug and go with what I’ve got. It’s a tough call. Ah, what the heck; besides, it’s been over a week since I did the last one of these.

Bill Amend’s FoxTrot (December 7) bids to get posted on mathematics teachers’ walls with a bit of play on two common uses of the term “degree”. It’s also natural to wonder why the same word “degree” should be used to represent the units of temperature and the size of an angle, to the point that they even use the same symbol of a tiny circle elevated from the baseline as a shorthand representation. As best I can make out, the use of the word degree traces back to Old French, and “degré”, meaning a step, as in a stair. In Middle English this got expanded to the notion of one of a hierarchy of steps, and if you consider the temperature of a thing, or the width of an angle, as something that can be grown or shrunk then … I’m left wondering if the Middle English folks who extended “degree” to temperatures and angles thought there were discrete steps by which either quantity could change.

As for the little degree symbol, Florian Cajori notes in A History Of Mathematical Notations that while the symbol (and the ‘ and ” for minutes and seconds) can be found in Ptolemy (!), in describing Babylonian sexagesimal fractions, this doesn’t directly lead to the modern symbols. Medieval manuscripts and early printed books would use abbreviations of Latin words describing what the numbers represented. Cajori rates as the first modern appearance of the degree symbol an appendix, composed by one Jacques Peletier, to the 1569 edition of the text Arithmeticae practicae methods facilis by Gemma Frisius (you remember him; the guy who made triangulation into something that could be used for surveying territories). Peletier was describing astronomical fractions, and used the symbol to denote that the thing before it was a whole number. By 1571 Erasmus Reinhold (whom you remember from working out the “Prutenic Tables”, updated astronomical charts that helped convince people of the use of the Copernican model of the solar system and advance the cause of calendar reform) was using the little circle to represent degrees, and Tycho Brahe followed his example, and soon … well, it took a century or so of competing symbols, including “Grad” or “Gr” or “G” to represent degree, but the little circle eventually won out. (Assume the story is more complicated than this. It always is.)

Mark Litzer’s Joe Vanilla (December 7) uses a panel of calculus to suggest something particularly deep or intellectually challenging. As it happens, the problem isn’t quite defined well enough to solve, but if you make a reasonable assumption about what’s meant, then it becomes easy to say: this expression is “some infinitely large number”. Here’s why.

The numerator is the integral \int_{0}^{\infty} e^{\pi} + \sin^2\left(x\right) dx . You can think of the integral of a positive-valued expression as the area underneath that expression and between the lines marked by, on the left, x = 0 (the number on the bottom of the integral sign), and on the right, x = \infty (the number on the top of the integral sign). (You know that it’s x because the integral symbol ends with “dx”; if it ended “dy” then the integral would tell you the left and the right bounds for the variable y instead.) Now, e^{\pi} + \sin^2\left(x\right) is a number that depends on x, yes, but which is never smaller than e^{\pi} (about 23.14) nor bigger than e^{\pi} + 1 (about 24.14). So the area underneath this expression has to be at least as big as the area within a rectangle that’s got a bottom edge at y = 0, a top edge at y = 23, a left edge at x = 0, and a right edge at x infinitely far off to the right. That rectangle’s got an infinitely large area. The area underneath this expression has to be no smaller than that.

Just because the numerator’s infinitely large doesn’t mean that the fraction is, though. It’s imaginable that the denominator is also infinitely large, and more wondrously, is large in a way that makes the ratio some more familiar number like “3”. Spoiler: it isn’t.

Actually, as it is, the denominator isn’t quite much of anything. It’s a summation; that’s what the capital sigma designates there. By convention, the summation symbol means to evaluate whatever expression there is to the right of it — in this case, it’s x^{\frac{1}{e}} + \cos\left(x\right) — for each of a series of values of some index variable. That variable is normally identified underneath the sigma, with a line such as x = 1, and (again by convention) for x = 2, x = 3, x = 4, and so on, until x equals whatever the number on top of the sigma is. In this case, the bottom doesn’t actually say what the index should be, although since “x” is the only thing that makes sense as a variable within the expression — “cos” means the cosine function, and “e” means the number that’s about 2.71828 unless it’s otherwise made explicit — we can suppose that this is a normal bit of shorthand like you use when context is clear.

With that assumption about what’s meant, then, we know the denominator is whatever number is represented by \left(1^{\frac{1}{e}} + \cos\left(1\right)\right) + \left(2^{\frac{1}{e}} + \cos\left(2\right)\right) + \left(3^{\frac{1}{e}} + \cos\left(3\right)\right) + \left(4^{\frac{1}{e}} + \cos\left(4\right)\right) +  \cdots + \left(10^{\frac{1}{e}} + \cos\left(10\right)\right) (and 1/e is about 0.368). That’s a number about 16.549, which falls short of being infinitely large by an infinitely large amount.

So, the original fraction shown represents an infinitely large number.

Mark Tatulli’s Lio (December 7) is another “anthropomorphic numbers” genre comic, and since it’s Lio the numbers naturally act a bit mischievously.

Greg Evans’s Luann Againn (December 7, I suppose technically a rerun) only has a bit of mathematical content, as it’s really playing more on short- and long-term memories. Normal people, it seems, have a buffer of something around eight numbers that they can remember without losing track of them, and it’s surprisingly easy to overload that. I recall reading, I think in Joseph T Hallinan’s Why We Make Mistakes: How We Look Without Seeing, Forget Things In Seconds, And Are All Pretty Sure We are Way Above Average, and don’t think I’m not aware of how funny it would be if I were getting this source wrong, that it’s possible to cheat a little bit on the size of one’s number-buffer.

Hallinan (?) gave the example of a runner who was able to remember strings of dozens of numbers, well past the norm, but apparently by the trick of parsing numbers into plausible running times. That is, the person would remember “834126120820” perfectly because it could be expressed as four numbers, “8:34, 1:26, 1:20, 8:20”, that might be credible running times for something or other and the runner was used to remembering such times. Supporting the idea that this trick was based on turning a lot of digits into a few small numbers was that the runner would be lost if the digits could not be parsed into a meaningful time, like, “489162693077”. So, in short, people are really weird in how they remember and don’t remember things.

Harley Schwadron’s 9 to 5 (December 8) is a “reluctant student” question who, in the tradition of kids in comic strips, tosses out the word “app” in the hopes of upgrading the action into a joke. I’m sympathetic to the kid not wanting to do long division. In arithmetic the way I was taught it, this was the first kind of problem where you pretty much had to approximate and make a guess what the answer might be and improve your guess from that starting point, and that’s a terrifying thing when, up to that point, arithmetic has been a series of predictable, discrete, universally applicable rules not requiring you to make a guess. It feels wasteful of effort to work out, say, what seven times your divisor is when it turns out it’ll go into the dividend eight times. I am glad that teaching approaches to arithmetic seem to be turning towards “make approximate or estimated answers, and try to improve those” as a general rule, since often taking your best guess and then improving it is the best way to get a good answer, not just in long division, and the less terrifying that move is, the better.

Justin Boyd’s Invisible Bread (December 12) reveals the joy and the potential menace of charts and graphs. It’s a reassuring red dot at the end of this graph of relevant-graph-probabilities.

Several comics chose to mention the coincidence of the 13th of December being (in the United States standard for shorthand dating) 12-13-14. Chip Sansom’s The Born Loser does the joke about how yes, this sequence won’t recur in (most of our) lives, but neither will any other. Stuart Carlson and Jerry Resler’s Gray Matters takes a little imprecision in calling it “the last date this century to have a consecutive pattern”, something the Grays, if the strip is still running, will realize on 1/2/34 at the latest. And Francesco Marciuliano’s Medium Large uses the neat pattern of the dates as a dip into numerology and the kinds of manias that staring too closely into neat patterns can encourage.

Reading The Comics, November 14, 2014: Rectangular States Edition


I have no idea why Comic Strip Master Command decided this week should see everybody do some mathematics-themed comic strips, but, so they did, and here’s my collection of the, I estimate, six hundred comic strips that touched on something recently. Good luck reading it all.

Samsons Dark Side of the Horse (November 10) is another entry on the theme of not answering the word problem.

Scott Adams’s Dilbert Classics (November 10) started a sequence in which Dilbert gets told the big boss was a geometry major, so, what can he say about rectangles? Further rumors indicate he’s more a geography fan, shifting Dilbert’s topic to the “many” rectangular states of the United States. Of course, there’s only two literally rectangular states, but — and Mark Stein’s How The States Got Their Shapes contains a lot of good explanations of this — many of the states are approximately rectangular. After all, when many of the state boundaries were laid out, the federal government had only vague if any idea what the landscapes looked like in detail, and there weren’t many existing indigenous boundaries the white governments cared about. So setting a proposed territory’s bounds to be within particular lines of latitude and longitude, with some modification for rivers or shorelines or mountain ranges known to exist, is easy, and can be done with rather little of the ambiguity or contradictory nonsense that plagued the eastern states (where, say, a colony’s boundary might be defined as where a river intersects a line of latitude that in fact it never touches). And while perfect rectangularity may be achieved only by Colorado and Wyoming, quite a few states — the Dakotas, Washington, Oregon, Missisippi, Alabama, Iowa — are rectangular enough.

Mikael Wulff and Anders Morgenthaler’s WuMo (November 10) shows that their interest in pi isn’t just a casual thing. They think about what those neglected and non-famous numbers get up to.

Sherman does poorly with mathematic problems that look more like short stories. And don't have pictures.
Jim Toomey’s Sherman’s Lagoon for the 11th of November, 2014. He’s got a point about pictures helping with this kind of problem.

Jim Toomey’s Sherman’s Lagoon starts a “struggling with mathematics homework” story on the 11th, with Sherman himself stumped by a problem that “looks more like a short story” than a math problem. By the 14th Megan points out that it’s a problem that really doesn’t make sense when applied to sharks. Such is the natural hazard in writing a perfectly good word problem without considering the audience.

Homework: Megan points out that a problem about walking in the rain with umbrellas doesn't make much sense for sharks or other underwater creatures.
Jim Toomey’s Sherman’s Lagoon for the 14th of November, 2014.

Mike Peters’s Mother Goose and Grimm (November 12) takes one of its (frequent) breaks from the title characters for a panel-strip-style gag about Roman numerals.

A Roman medic calls for an IV; the nurse asks if he means this numeral 4.
Mike Peters’s Mother Goose and Grimm for the 12th of November, 2014.

Darrin Bell’s Candorville (November 12) starts talking about Zeno’s paradox — not the first time this month that a comic strip’s gotten to the apparent problem of covering any distance when distance is infinitely divisible. On November 13th it’s extended to covering stretches of time, which has exactly the same problem. Now it’s worth reminding people, because a stunning number of them don’t seem to understand this, that Zeno was not suggesting that there’s no such thing as motion (or that he couldn’t imagine an infinite convergent sequence; it’s easy to think of a geometric construction that would satisfy any ancient geometer); he was pointing out that there’s things that don’t make perfect sense about it. Either distance (and time) are infinitely divisible into indistinguishable units, or they are not; and either way has implications that seem contrary to the way motion works. Perhaps they can be rationalized; perhaps they can’t; but when you can find a question that’s easy to pose and hard to answer, you’re probably looking at something really worth thinking hard about.

Bill Amend’s FoxTrot Classics (November 12, a rerun) puns on the various meanings of “irrational”. A fun little fact you might want to try proving sometime, though I wouldn’t fault you if you only tried it out for a couple specific numbers and decided the general case too much to do: any whole number — like 2, 3, 4, or so on — has a square root that’s either another whole number, or else has a square root that’s irrational. There’s not a case where, say, the square root is exactly 45.144 or something like that, though it might be close.

Susan tries to figure out either what year she was in Grade Ten or what her age was back then. She admits she was never any good at math, although the real trouble might be she hasn't got a clear idea what she wants to calculate.
Sandra Bell-Lundy’s Between Friends for the 13th of November, 2014.

Sandra Bell-Lundy’sBetween Friends (November 13) shows one of those cases where mental arithmetic really is useful, as Susan tries to work out — actually, staring at it, I’m not precisely sure what she is trying to work out. Her and her coffee partner’s ages in Grade Ten, probably, or else just when Grade Ten was. That’s most likely her real problem: if you don’t know what you’re looking for it’s very difficult to find it. Don’t start calculating before you know what you’re trying to work out.

If I wanted to work out what year was 35 years ago I’d probably just use a hack: 35 years before 2014 is one year before “35 years before 2015”, which is a much easier problem to do. 35 years before 2015 is also 20 years before 2000, which is 1980, so subtract one and you get 1979. (Alternatively, I might remember it was 35 years ago that the Buggles’ “Video Killed The Radio Star” first appeared, which I admit is not a method that would work for everyone, or for all years.) If I wanted to work out my (and my partner’s) age in Grade Ten … well, I’d use a slightly different hack: I remember very well that I was ten years old in Grade Five (seriously, the fact that twice my grade was my age overwhelmed my thinking on my tenth birthday, which is probably why I had to stay in mathematics), so, add five to that and I’d be 15 in Grade Ten.

Bill Whitehead’s Free Range (November 13) brings up one of the most-quoted equations in the world in order to show off how kids will insult each other, which is fair enough.

Rick Detorie’s One Big Happy (November 13), this one a rerun from a couple years ago because that’s how his strip works on Gocomics, goes to one of its regular bits of the kid Ruthie teaching anyone she can get in range, and while there’s a bit more to arithmetic than just adding two numbers to get a bigger number, she is showing off an understanding of a useful sanity check: if you add together two (positive) numbers, you have to get a result that’s bigger than either of the ones you started with. As for the 14th, and counting higher, well, there’s not much she could do about that.

Steve McGarry’s Badlands (November 14) talks about the kind of problem people wish to have: how to win a lottery where nobody else picks the same numbers, so that the prize goes undivided? The answer, of course, is to have a set of numbers that nobody else picked, but is there any way to guarantee that? And this gets into the curious psychology of random numbers: there is absolutely no reason that 1-2-3-4-5-6, or for that matter 7-8-9-10-11-12, would not come up just as often as, say, 11-37-39-51-52-55, but the latter set looks more random. But we see some strings of numbers as obviously a pattern, while others we don’t see, and we tend to confuse “we don’t know the pattern” with “there is no pattern”. I have heard the lore that actually a disproportionate number of people pick such obvious patterns like 1-2-3-4-5-6, or numbers that form neat pictures on a lottery card, no doubt cackling at how much more clever they are than the average person, and guaranteeing that if such a string ever does come out there’ll a large number of very surprised lottery winners. All silliness, really; the thing to do, obviously, is buy two tickets with the exact same set of numbers, so that if you do win, you get twice the share of anyone else, unless they’ve figured out the same trick.

Reading the Comics, October 7, 2014: Repeated Comics Edition


Since my last roundup of mathematics-themed comic strips there’s been a modest drizzle of new ones, and I’m not sure that I can find any particular themes to them, except that Zach Weinersmith and the artistic collective behind Eric the Circle apparently like my attention. Well, what the heck; that’s easy enough to give.

Zach Weinersmith’s Saturday Morning Breakfast Cereal (September 29) hopes to be that guy who appears somewhere around the fourth comment of every news article ever that mentions a correlation being found between two quantities. A lot of what’s valuable about science is finding causal links between things, but it’s only in rare and, often, rather artificial circumstances that such links are easy to show. What’s more often necessary is showing that as one quantity changes so does another, which allows one to suspect a link. Then, typically, one would look for a plausible reason they might have anything to do with one another, and look for ways to experiment and prove whether there is or is not.

But just because there is a correlation doesn’t by itself mean that one thing necessarily has anything to do with another. They could be coincidence, for example, or they could be influenced by some other confounding factor. To be worth mention in a decent journal, a correlation is probably going to be strong enough that it’s hard to believe it’s just coincidence, but there could yet be some confounding factor. And even if there is a causal link, in the complicated mess that is reality it can be difficult to discern which way the link flows. This is summarized in deductive logic by saying that correlation does not imply causation, but that uses deductive logic’s definition of “imply”.

In deductive logic to say “this implies that” means it is impossible for “this” to be true and “that” false simultaneously. It is perfectly permissible for both “this” and “that” to be true, and permissible for “this” to be false and “that” false, and — this is the point where Intro to Logic students typically crash — permissible for “this” to be false and “that” true. Colloquially, though, “imply” has a different connotation, something more along the lines of “this” and “that” have to both be false or both be true together. Don’t make that mistake on your logic test.

When a logician says that correlation does not imply causation, she is saying that it is imaginable for the correlation to be true while the causation is false. She is not saying the causation is false; she is just saying that the case is not proved from the fact of a correlation being true. And that’s so; if we just knew two things were correlated we would have to experiment to find whether there is a causal link. But finding a correlation one of the ways to start finding casual links; it’d be obviously daft not to use them as the start of one’s search. Anyway, that guy in about the fourth comment of every news report about a correlation just wants you to know it’s very important he tell you he’s smarter than journalists.

Saturday Morning Breakfast Cereal pops back up again (October 1) with an easier-to-describe joke about August Ferdinand Möbius and his rather famous strip, here applied to the old gag about walking to school uphill both ways. One hates to be a spoilsport, but Möbius was educated at home until 13, so this comic is not reliable as a shorthand biography of the renowned mathematician.

Eric the Circle has had a couple strips by “Griffinetsabine”, one on October 3, and another on the 7th of October, based on the Shape Singles Bar. Both strips are jokes about two points connecting by a line, suggesting that Griffinetsabine knew the premise was good for a couple of variants. I’d have spaced out the publication of them farther but perhaps this was the best that could be done.

Mikael Wulff and Anders Morgenthaler’s Truth Facts (September 30) — a panel strip that’s often engaging in showing comic charts — gives a guide to what the number of digits you’ve memorized says about you. (For what it’s worth, I peter out at “897932”.) I’m mildly delighted to find that their marker for Isaac Newton is more or less correct: Newton did work out pi to fifteen decimal places, by using his binomial theorem and a calculation of the area within a particular wedge of the circle. (As I make it out Wulff and Morgenthaler put Newton at fourteen decimal points, but they might have read references to Newton working out “fifteen decimal points” as meaning something different to what I do.) Newton’s was not the best calculation of pi in the 1660s when he worked it out — Christoph Grienberger, an Austrian Jesuit astronomer, had calculated 38 decimal places a generation earlier — but I can’t blame Wulff and Morgenthaler for supposing Newton to be a more recognizable name than Grienberger. I imagine if Einstein or Stephen Hawking had done any particularly unique work in calculating the digits of pi they’d have appeared on the chart too.

John Graziano’s Ripley’s Believe It or Not (October 1) — and don’t tell me that attribution doesn’t look weird — shares a story about the followers of the Ancient Greek mathematician, philosopher, and mystic Pythagoras, that they were forbidden to wear wool, eat beans, or pick up things they had dropped. I have heard the beans thing before and I think I’ve heard the wool prohibition before, but I don’t remember hearing about them not being able to pick up things before.

I’m not sure I can believe it, though: Pythagoras was a strange fellow, so far as the historical record is clear. It’s hard to be sure just what is true about him and his followers, though, and what is made up, either out of devoted followers building up the figure they admire or out of critics making fun of a strange fellow with his own little cult. Perhaps it’s so, perhaps it’s not. I would like to see a primary source, and I don’t think any exist.

The Little King skates a figure 8 that requires less tricky curving.
Otto Soglow’s The Little King for 29 February 1948.

Otto Soglow’s The Little King (October 5; originally run February 29, 1948) provides its normal gentle, genial humor in the Little King working his way around the problem of doing a figure 8.

Reading the Comics, July 24, 2014: Math Is Just Hard Stuff, Right? Edition


Maybe there is no pattern to how Comic Strip Master Command directs the making of mathematics-themed comic strips. It hasn’t quite been a week since I had enough to gather up again. But it’s clearly the summertime anyway; the most common theme this time seems to be just that mathematics is some hard stuff, without digging much into particular subjects. I can work with that.

Pab Sungenis’s The New Adventures of Queen Victoria (July 19) brings in Erwin Schrödinger and his in-strip cat Barfly for a knock-knock joke about proof, with Andrew Wiles’s name dropped probably because he’s the only person who’s gotten to be famous for a mathematical proof. Wiles certainly deserves fame for proving Fermat’s Last Theorem and opening up what I understand to be a useful new field for mathematical research (Fermat’s Last Theorem by itself is nice but unimportant; the tools developed to prove it, though, that’s worthwhile), but remembering only Wiles does slight Richard Taylor, whose help Wiles needed to close a flaw in his proof.

Incidentally I don’t know why the cat is named Barfly. It has the feel to me of a name that was a punchline for one strip and then Sungenis felt stuck with it. As Thomas Dye of the web comic Newshounds said, “Joke names’ll kill you”. (I’m inclined to think that funny names can work, as the Marx Brotehrs, Fred Allen, and Vic and Sade did well with them, but they have to be a less demanding kind of funny.)

John Deering’s Strange Brew (July 19) uses a panel full of mathematical symbols scrawled out as the representation of “this is something really hard being worked out”. I suppose this one could also be filed under “rocket science themed comics”, but it comes from almost the first problem of mathematical physics: if you shoot something straight up, how long will it take to fall back down? The faster the thing starts up, the longer it takes to fall back, until at some speed — the escape velocity — it never comes back. This is because the size of the gravitational attraction between two things decreases as they get farther apart. At or above the escape velocity, the thing has enough speed that all the pulling of gravity, from the planet or moon or whatever you’re escaping from, will not suffice to slow the thing down to a stop and make it fall back down.

The escape velocity depends on the size of the planet or moon or sun or galaxy or whatever you’re escaping from, of course, and how close to the surface (or center) you start from. It also assumes you’re talking about the speed when the thing starts flying away, that is, that the thing doesn’t fire rockets or get a speed boost by flying past another planet or anything like that. And things don’t have to reach the escape velocity to be useful. Nothing that’s in earth orbit has reached the earth’s escape velocity, for example. I suppose that last case is akin to how you can still get some stuff done without getting out of the recliner.

Mel Henze’s Gentle Creatures (July 21) uses mathematics as the standard for proving intelligence exists. I’ve got a vested interest in supporting that proposition, but I can’t bring myself to say more than that it shows a particular kind of intelligence exists. I appreciate the equation of the final panel, though, as it can be pretty well generalized.

To disguise a sports venue it's labelled ``Math Arena'', with ``lectures on the actual odds of beating the casino''.
Bill Holbrook’s _Safe Havens_ for the 22nd of July, 2014.

Bill Holbrook’s Safe Havens (July 22) plays on mathematics’ reputation of being not very much a crowd-pleasing activity. That’s all right, although I think Holbrook makes a mistake by having the arena claim to offer a “lecture on the actual odds of beating the casino”, since the mathematics of gambling is just the sort of mathematics I think would draw a crowd. Probability enjoys a particular sweet spot for popular treatment: many problems don’t require great amounts of background to understand, and have results that are surprising, but which have reasons that are easy to follow and don’t require sophisticated arguments, and are about problems that are easy to imagine or easy to find interesting: cards being drawn, dice being rolled, coincidences being found, or secrets being revealed. I understand Holbrook’s editorial cartoon-type point behind the lecture notice he put up, but the venue would have better scared off audiences if it offered a lecture on, say, “Chromatic polynomials for rigidly achiral graphs: new work on Yamada’s invariant”. I’m not sure I could even explain that title in 1200 words.

Missy Meyer’s Holiday Doodles (July 22) revelas to me that apparently the 22nd of July was “Casual Pi Day”. Yeah, I suppose that passes. I didn’t see much about it in my Twitter feed, but maybe I need some more acquaintances who don’t write dates American-fashion.

Thom Bluemel’s Birdbrains (July 24) again uses mathematics — particularly, Calculus — as not just the marker for intelligence but also as The Thing which will decide whether a kid goes on to success in life. I think the dolphin (I guess it’s a dolphin?) parent is being particularly horrible here, as it’s not as if a “B+” is in any way a grade to be ashamed of, and telling kids it is either drives them to give up on caring about grades, or makes them send whiny e-mails to their instructors about how they need this grade and don’t understand why they can’t just do some make-up work for it. Anyway, it makes the kid miserable, it makes the kid’s teachers or professors miserable, and for crying out loud, it’s a B+.

(I’m also not sure whether a dolphin would consider a career at Sea World success in life, but that’s a separate and very sad issue.)

Reading the Comics, July 18, 2014: Summer Doldrums Edition


Now, there, see? The school year (in the United States) has let out for summer and the rush of mathematics-themed comic strips has subsided; it’s been over two weeks since the last bunch was big enough. Given enough time, though, a handful of comics will assemble that I can do something with, anything, and now’s that time. I hate to admit also that they’re clearly not trying very hard with these mathematics comics as they’re not about very juicy topics. Call it the summer doldroms, as I did.

Mason Mastroianni and Mick Mastroianni’s B.C. (July 6) spends most of its text talking about learning cursive, as part of a joke built around the punch line that gadgets are spoiling students who learn to depend on them instead of their own minds. So it would naturally get around to using calculators (or calculator apps, which is a fair enough substitute) in place of mathematics lessons. I confess I come down on the side that wonders why it’s necessary to do more than rough, approximate arithmetic calculations without a tool, and isn’t sure exactly what’s gained by learning cursive handwriting, but these are subjects that inspire heated and ongoing debates so you’ll never catch me admitting either position in public.

Eric the Circle (July 7), here by “andel”, shows what one commenter correctly identifies as a “pi fight”, which might have made a better caption for the strip, at least for me, because Eric’s string of digits wasn’t one of the approximations to pi that I was familiar with. I still can’t find it, actually, and wonder if andel didn’t just get a digit wrong. (I might just not have found a good web page that lists the digits of various approximations to pi, I admit.) Erica’s approximation is the rather famous 22/7.

Richard Thompson’s Richard’s Poor Almanac (July 7, rerun) brings back our favorite set of infinite monkeys, here, to discuss their ambitious book set at the Museum of Natural History.

Tom Thaves’s Frank and Ernest (July 16) builds on the (true) point that the ancient Greeks had no symbol for zero, and would probably have had a fair number of objections to the concept.

'The day Einstein got the wind knocked out of his sails': Einstein tells his wife he's discovered the theory of relativity.
Joe Martin’s _Mr Boffo_ strip for the 18th of July, 2014.

Joe Martin’s Mr Boffo (July 18, sorry that I can’t find a truly permanent link) plays with one of Martin’s favorite themes, putting deep domesticity to great inventors and great minds. I suspect but do not know that Martin was aware that Einstein’s first wife, Mileva Maric, was a fellow student with him at the Swiss Federal Polytechnic. She studied mathematics and physics. The extent to which she helped Einstein develop his theories is debatable; as far as I’m aware the evidence only goes so far as to prove she was a bright, outside mind who could intelligently discuss whatever he might be wrangling over. This shouldn’t be minimized: describing a problem is often a key step in working through it, and a person who can ask good follow-up questions about a problem is invaluable even if that person doesn’t do anything further.

Charles Schulz’s Peanuts (July 18) — a rerun, of course, from the 21st of July, 1967 — mentions Sally going to Summer School and learning all about the astronomical details of summertime. Astronomy has always been one of the things driving mathematical discovery, but I admit, thinking mostly this would be a good chance to point out Dr Helmer Aslaksen’s page describing the relationship between the solstices and the times of earliest and latest sunrise (and sunset). It’s not quite as easy as finding when the days are longest and shortest. Dr Aslaksen has a number of fascinating astronomy- and calendar-based pages which I think worth reading, so, I hope you enjoy.

Reading the Comics, June 4, 2014: Intro Algebra Edition


I’m not sure that there is a theme to the most recent mathematically-themed comic strips that I’ve seen, all from GoComics in the past week, but they put me in mind of the stuff encountered in learning algebra, so let’s run with that. It’s either that or I start making these “edition” titles into something absolutely and utterly meaningless, which could be.

Marc Anderson’s Andertoons (May 30) uses the classic setup of a board full of equation to indicate some serious, advanced thinking going on, and then puts in a cute animal twist on things. I don’t believe that the equation signifies anything, but I have to admit I’m not sure. It looks quite plausibly like something which might turn up in quantum mechanics (the “h” and “c” and lambda are awfully suggestive), so if Anderson made it up out of whole cloth he did an admirable job. If he didn’t make it up and someone recognizes it, please, let me know; I’m curious what it might be.

Marc Anderson reappears on the second of June has the classic reluctant student upset with the teacher who knew all along what x was. Knowledge of what x is is probably the source of most jokes about learning algebra, or maybe mathematics overall, and it’s amusing to me anyway that what we really care about is not what x is particularly — we don’t even do ourselves any harm if we call it some other letter, or for that matter an empty box — but learning how to figure out what values in the place of x would make the relationship true.

Jonathan Lemon’s Rabbits Against Magic (May 31) has the one-eyed rabbit Weenus doing miscellaneous arithmetic on the way to punning about things working out. I suppose to get to that punch line you have to either have mathematics or gym class as the topic, and I wouldn’t be surprised if Lemon’s done a version with those weight-lifting machines on screen. That’s not because I doubt his creativity, just that it’s the logical setup.

Eric Scott’s Back In The Day (June 2) has a pair of dinosaurs wondering about how many stars there are. Astronomy has always inspired mathematics. After one counts the number of stars one gets to wondering, how big the universe could be — Archimedes, classically, estimated the universe was about big enough to hold 1063 grains of sand — or how far away the sun might be — which the Ancient Greeks were able to estimate to the right order of magnitude on geometric grounds — and I imagine that looking deep into the sky can inspire the idea that the infinitely large and the infinitely small are at least things we can try to understand. Trying to count stars is a good start.

Steve Boreman’s Little Dog Lost (June 2) has a stick insect provide the excuse for some geometry puns.

Brian and Ron Boychuk’s The Chuckle Brothers (June 4) has a pie shop gag that I bet the Boychuks are kicking themselves for not having published back in mid-March.

Reading The Comics, March 17, 2014: After The Ides Edition


Rather than wait to read today’s comics I’m just going to put in a fresh entry going over mathematical points raised in the funny pages. This one turned out to include a massive diversion into the wonders of the ancient Roman calendar, which is a mathematical topic, really, although there’s no calculations involved in it just here.

Bill Hinds’s Cleats (March 7, rerun) calls on one of the common cultural references to percentages, the idea of athletes giving 100 percent efforts. (Edith is feeling more like an 80 percent effort, or less than that.) The idea of giving 100 percent in a sport is one that invites the question, 100 percent of what; granting that there is some standard expectable effort made, then, even the sports reporting cliche of giving 110 percent is meaningful.
Cleats continued on the theme the next day, as Edith was thinking more of giving about 79 percent of 80 percent, and it’s not actually that hard to work out in your head what percent that is, if you know anything about doing arithmetic in your head.

Jef Mallet’s Frazz (March 14) was not actually the only comic strip among the roster I normally read to make a Pi Day reference, but I think it suffices as the example for the whole breed. I admit that I feel a bit curmudgeonly that I don’t actually care about Pi Day. I suppose that as a chance for people to promote the idea of learning mathematics, and maybe attach it to some of the many interesting things to be said about mathematics using Pi as the introductory note the idea is fine, but just naming a thing isn’t by itself a joke. I’m told that Facebook (I’m not on it) was thick with people posting photographs of pies, which is probably more fun when you think of it than when you notice everybody else thought of it too. Anyway, organized Pi Day events are still pretty new as Internet Pop Holidays go. Perhaps next year’s comics will be sharper.

Jenny Campbell’s Flo and Friends (March 15) comes back to useful mental arithmetic work, in this case in working out a reasonable tip. A twenty-percent tip is, mercifully, pretty easy to remember just as what’s-her-name specifies. (I can’t think of the kid’s name and there’s no meet-our-cast page on the web site. None of the commenters mention her name either, although they do make room to insult health care reform and letting students use calculators to do arithmetic, so, I’m sorry I read that far down too.) But as ever you need to make sure the process is explained clearly and understood, and Tina needed to run a sanity check on the result. Sanity checks, as suggested, won’t show that your answer is right, but they will rule out some of the wrong ones. (A fifteen percent tip is a bit annoying to calculate exactly, but dividing the original amount by six will give you a sixteen-and-two-thirds percent tip, which is surely close enough, especially if you round off to a quarter-dollar.)

Steve Breen and Mike Thompson’s Grand Avenue (March 15) has the kids wonder what are the ides of March; besides that they’re the 15th of the month and they’re used for some memorable writing about Julius Caesar it’s a fair thing not to know. They derive from calendar-keeping, one of the oldest useful applications of mathematics and astronomy. The ancient Roman scheme set three special dates in the month: the kalends, which seem to have started as the day of the new moon as observed in Rome; the nones, when the moon was at its first quarter; and the ides, when the moon was full.

But by the time of Numa Pompilius, the second (traditional) King of Rome, who reformed the calendar around 713 BC, the lunar link was snapped, partly so that the calendar year could more nearly fit the length of the time it takes to go from one spring to another. (Among other things the pre-Numa calendar had only ten months, with the days between December and March not belonging to any month; since Romans were rather agricultural at the time and there wasn’t much happening in winter, this wasn’t really absurd, even if I find it hard to imagine living by this sort of standard. After Numa there were only about eleven days of the year unaccounted for, with the time made up, when it needed to be, by inserting an extra month, Mercedonius, in the middle of February.) Months then had, February excepted, either 29 or 31 days, with the ides being on the fifteenth day of the 31-day months (March, May, July, and October) and the thirteenth day of the 29-day months.

For reasons that surely made sense if you were an ancient Roman the day was specified as the number of days until the next kalend, none, or ide; so, for example, while the 13th of March would be the 2nd day before the ides of March, II Id Mar, the 19th of March would be recorded as the the the 14th day before the kalend of April, or, XIV Kal Apr. I admit I could probably warm up to counting down to the next month event, but the idea of having half the month of March written down on the calendar as a date with “April” in it leaves me deeply unsettled. And that’s before we even get into how an extra month might get slipped into the middle of February (between the 23rd and the 24th of the month, the trace of which can still be observed in the dominical letters of February in leap years, on Roman Catholic and Anglican calendars, and in the obscure term “bissextile year” for leap year). But now that you see that, you know why (a) the ancient Romans had so much trouble getting their database software to do dates correctly and (b) you get to be all smugly superior to anyone who tries making a crack about the United States Federal Income Tax deadline being on the Ides of April, since they never are.

(Warning: absolutely no one ever will be impressed by your knowledge of the Ides of April and their inapplicability to discussions of the United States Federal Income Tax. However, you might use this as a way to appear like you’re making friendly small talk while actually encouraging people to leave you alone.)

Tom Horacek’s Foolish Mortals (March 17), an erratically-published panel strip, calls on the legend of how mathematicians “usually” peak in their twenties. It’s certainly said of mathematicians that they do their most important work while young — note that the Fields Medal is explicitly given to mathematicians for work done when they were under forty years old — although I’m not aware of anyone who’s actually studied this, and the number of great mathematicians who insist on doing brilliant work into their old age is pretty impressive.

Certainly, for example, Newton began work on calculus (and optics and gravitation) when he was about 23, but he didn’t publish until he was about fifty. (Leibniz, meanwhile, started publishing calculus his way at about age 38.) It’s probably impossible to say what Leonhard Euler’s most important work was, but (for example) his equations describing inviscid fluids — which would be the masterpiece for anybody not Euler — he published when he was fifty. Carl Friedrich Gauss didn’t start serious work in electromagnetism until he was about 55 years old, too. The law of electric flux which Gauss worked out for that — which, again, would have been the career achievement if Gauss weren’t overflowing with them — he published when he was 58.

I guess that I’m saying is that great minds, at least, don’t necessarily peak in their twenties, or at least they have some impressive peaks afterwards too.

Reading the Comics, February 11, 2014: Running Out Pi Edition


I’d figured I had enough mathematics comic strips for another of these entries, and discovered during the writing that I had much more to say about one than I had anticipated. So, although it’s no longer quite the 11th, or close to it, I’m going to exile the comics from after that date to the next of these entries.

Melissa DeJesus and Ed Power’s My Cage (February 6, rerun) makes another reference to the infinite-monkeys-with-typewriters scenario, which, since it takes place in a furry universe allows access to the punchline you might expect. I’ve written about that before, as the infinite monkeys problem sits at a wonderful intersection of important mathematics and captivating metaphors.

Gene Weingarten, Dan Weingarten, and David Clark’s Barney and Clyde (starting February 10) (and when am I going to make a macro for that credit and title?) has Cynthia given a slightly baffling homework lesson: to calculate the first ten digits of pi. The story continues through the 11th, the 12th, the 13th, finally resolving on the the 14th, in the way such stories must. I admit I’m not sure why exactly calculating the digits of π would be a suitable homework assignment; I can see working out division problems until the numbers start repeating, or doing a square root or something by hand until you’ve found enough digits.

π, though … well, there’s the question of why it’d be an assignment to start with, but also, what formula for generating π could be plausibly appropriate for an elementary school class. The one that seems obvious to me — π is equal to four times (1/1 minus 1/3 plus 1/5 minus 1/7 plus 1/9 minus 1/11 and so on and so on) — also takes way too long to work. If a little bit of coding is right, it takes something like 160 terms to get just the first two digits of π correct and that isn’t even stable. (The first 160 terms add to 3.135; the first 161 terms to 3.147.) Getting it to ten digits would take —

Well, I thought it might be as few was 10,000 terms, because it turns out the sum of the first ten thousand terms in that series is 3.1414926536, which looks dead-on until you notice that π is 3.1415926536. That’s a neat coincidence, though.

Anyway, obviously, that formula wouldn’t do, and we see on the strip of the 14th that Lucretia isn’t using that. There are a great many formulas that generate the value of π, any of which might be used for a project like this; some of them get the digits right quite rapidly, usually at a cost of being very complicated. The formula shown in the strip of the 14th, though, doesn’t seem to be right. Lucretia’s work uses the formula \pi = \sqrt{12} \cdot \sum_{k = 0}^{\infty} \frac{(-3)^{-k}}{2k + 1} , which takes only about 21 terms to get to the demanded ten digits of accuracy. I don’t want to guess how many pages of work it would take to get to 13,908 places.

If I don’t miss my guess the formula used here is one by Abraham Sharp, an astronomer and mathematician who worked for the Royal Observatory at Greenwich and set a record by calculating π to 72 decimal digits. He was also an instrument-maker, of rather some skill, and I found a page purporting to show his notes of how to cut some complicated polyhedrons out of a block of wood, so, if my father wants to carve a 120-sided figure, here’s his chance. Sharp seems to have started with Leibniz’s formula (yes, that Leibniz) — that the arctangent of a number x is equal to x minus one-third x cubed plus one-fifth x to the fifth power minus one-seventh x to the seventh power, et cetera — with the knowledge that the arctangent of the square root of one-third is equal to one-sixth π and produced this series that looks a lot like the one we started with, but which gets digits correct so very much more quickly.

Darrin Bell’s Candorville (February 13) is primarily a bit of guys insulting friends, but what do you know and π makes a cameo appearance here.

Shannon Wheeler’s Too Much Coffee Man (February 10) is a Venn Diagram cartoon in the service of arguing that Venn Diagram cartoons aren’t funny. Putting aside the smoke and sparks popping out of the Nomad space probe which Kirk and Spock are rushing to the transporter room, I don’t think it’s quite fair: the ease the Venn diagram gives to grouping together concepts and showing how they relate helps organize one’s understanding of concepts and can be a really efficient way to set up a joke. Granting that, perhaps Wheeler’s seen too many Venn Diagram cartoons that fail, a complaint I’m sympathetic to.

Bill Amend’s FoxTrot (February 11, rerun) was one of those strips trying to be taped to the math teacher’s door, with the pun-based programming for the Math Channel.

Reading the Comics, February 1, 2014


For today’s round of mathematics-themed comic strips a little deeper pattern turns out to have emerged: π, that most popular of the transcendental numbers, turns up quite a bit in the comics that drew my attention the past couple weeks. Let me explain.

Dan Thompson’s Brevity (January 23) returns to the anthropomorphic numbers racket, with the kind of mathematics puns designed to get the strip pasted to the walls of the teacher’s lounge. I wonder how that’s going for him.

Greg Evans’s Luann Againn (January 25, rerun from 1986) has Luann not understanding how to work out an arithmetic problem until it’s shown how to do it: use the calculator. This is a joke that’s probably going to be with us as long as there are practical, personal calculating devices, because it is a good question why someone should bother learning arithmetic when a device will do it faster and better by every reasonable measure. I admit not being sure there is much point to learning arithmetic, other than as a way to practice a particular way of learning how to apply algorithms. I suppose it also stands as a way to get people who are really into mathematics to highlight themselves: someone who memorizes the times tables is probably interested in the kinds of systematic thought that mathematics depends on. But that’s a weak reason to demand it of every student. I suppose arithmetic is very testable, but that’s an even worse reason to make students go through it.

Mind you, I am quite open to the idea that arithmetic drills are useful for students. That I don’t know a particular reason why I should care whether a seventh-grader can divide 391 by 17 by hand doesn’t mean that I don’t think there is one.

Continue reading “Reading the Comics, February 1, 2014”

Reading the Comics, September 11, 2013


I may need to revise my seven-or-so-comic standard for hosting one of these roundups of mathematics-themed comic strips, at least during the summer vacation. We’ll see how they go as the school year picks up and cartoonists return to the traditional jokes of students not caring about algebra and kids giving wiseacre responses to word problems.

Jan Eliot’s Stone Soup began a sequence on the 26th of August in which Holly, the teenager, has to do flash cards to improve her memorization of the multiplication tables. It’s a baffling sequence to me, at least, since I can’t figure why a high schooler needs to study the times tables (on the 27th, Grandmom says it’s because it will make mathematics easier the more arithmetic she can do in her head). It’s also a bit infuriating because I can’t see a way to make sure Holly sees mathematics as tedious drudge work more than getting drilled by flash cards through summer vacation, particularly as she’s at an age where she ought to be doing algebra or trigonometry or geometry.

Steve Moore’s In The Bleachers (September 1) uses a bit of mathematics as a throwaway “something complicated to be thinking about” bit. I do like that the mathematics shown at least parses. I’m not sure offhand what problem the pitcher is trying to solve, that is, but the steps in it are done correctly, and even show off a nice bit of implicit differentiation. That’s a bit of differential calculus where you’ll find the rate of change of one variable with respect to another depends on the value of the variable, which isn’t actually hard to do if you follow the rules correctly but which, as I remember it, produces a vague sense of unease at its introduction. Probably it feels vaguely illicit to have a function defined in, in parts, in terms of itself.

Continue reading “Reading the Comics, September 11, 2013”

Quick Little Calculus Puzzle


fluffy, one of my friends and regular readers, got to discussing with me a couple of limit problems, particularly, ones that seemed to be solved through L’Hopital’s Rule and then ran across some that don’t call for that tool of Freshman Calculus which you maybe remember. It’s the thing about limits of zero divided by zero, or infinity divided by infinity. (It can also be applied to a couple of other “indeterminate forms”; I remember when I took this level calculus the teacher explaining there were seven such forms. Without looking them up, I think they’re \frac00, \frac{\infty}{\infty}, 0^0, \infty^{0}, 0^{\infty}, 1^{\infty}, \mbox{ and } \infty - \infty but I would not recommend trusting my memory in favor of actually studying for your test.)

Anyway, fluffy put forth two cute little puzzles that I had immediate responses for, and then started getting plagued by doubts about, so I thought I’d put them out here for people who want the recreation. They’re both about taking the limit at zero of fractions, specifically:

\lim_{x \rightarrow 0} \frac{e^x}{x^e}

\lim_{x \rightarrow 0} \frac{x^e}{e^x}

where e here is the base of the natural logarithm, that is, that number just a little high of 2.71828 that mathematicians find so interesting even though it isn’t pi.

The limit is, if you want to be exact, a subtly and carefully defined idea that took centuries of really bright work to explain. But the first really good feeling that I really got for it is to imagine a function evaluated at the points near but not exactly at the target point — in the limits here, where x equals zero — and to see, if you keep evaluating x very near zero, are the values of your expression very near something? If it does, that thing the expression gets near is probably the limit at that point.

So, yes, you can plug in values of x like 0.1 and 0.01 and 0.0001 and so on into \frac{e^x}{x^e} and \frac{x^e}{e^x} and get a feeling for what the limit probably is. Saying what it definitely is takes a little more work.

Going Fishing In Pi


I have to imagine I’ve run across this before, but here’s a nice little page which allows one to search the (base ten) digits of π for any sequence of up to 120 digits that one wants. It searches the first 200 million digits of pi, which is enough digits that you can be reasonably sure that any string of six or seven digits you look for are there, and it’s not ridiculously unlikely that a string of ten digits in a row will turn up. The natural question is, why is this interesting?

People who’ve learned a bit about pi may have heard that it’s probably a “normal number”, that is, a number whose digits contain every possible finite string of digits within it somewhere. That suggests that finding any particular string of digits in pi is no more surprising than finding any particular word in a complete dictionary (if we imagine there’s a dictionary that ever did include all the words of a language). The story’s a little more complicated than that.

Continue reading “Going Fishing In Pi”

How Many Numbers Have We Named?


I want to talk about some numbers which have names, and to argue that surprisingly few of numbers do. To make that argument it would be useful to say what numbers I think have names, and which ones haven’t; perhaps if I say enough I will find out.

For example, “one” is certainly a name of a number. So are “two” and “three” and so on, and going up to “twenty”, and going down to “zero”. But is “twenty-one” the name of a number, or just a label for the number described by the formula “take the number called twenty and add to it the number called one”?

It feels to me more like a label. I note for support the former London-dialect preference for writing such numbers as one-and-twenty, two-and-twenty, and so on, a construction still remembered in Charles Dickens, in nursery rhymes about blackbirds baked in pies, in poetry about the ways of constructing tribal lays correctly. It tells you how to calculate the number based on a few named numbers and some operations.

None of these are negative numbers. I can’t think of a properly named negative number, just ones we specify by prepending “minus” or “negative” to the label given a positive number. But negative numbers are fairly new things, a concept we have found comfortable for only a few centuries. Perhaps we will find something that simply must be named.

That tips my attitude (for today) about these names, that I admit “thirty” and “forty” and so up to a “hundred” as names. After that we return to what feel like formulas: a hundred and one, a hundred and ten, two hundred and fifty. We name a number, to say how many hundreds there are, and then whatever is left over. In ruling “thirty” in as a name and “three hundred” out I am being inconsistent; fortunately, I am speaking of peculiarities of the English language, so no one will notice. My dictionary notes the “-ty” suffix, going back to old English, means “groups of ten”. This makes “thirty” just “three tens”, stuffed down a little, yet somehow I think of “thirty” as different from “three hundred”, possibly because the latter does not appear in my dictionary. Somehow the impression formed in my mind before I thought to look.
Continue reading “How Many Numbers Have We Named?”