Reading the Comics, August 5, 2022: Catching Up Edition

I’ve had several weeks since my last Reading the Comics post. They’ve been quiet enough weeks. Let me share some of the recent offerings from Comic Strip Master Command that I enjoyed, though. I enjoy many comic strips but not all of them mention something on-point here.

Isabella Bannerman’s Six Chix for the 18th of July is a wordplay joke, naming a dog Isosceles to respect his front and hindlegs being equal in length. As I say, I’m including these because I like them, not because I have deep thoughts about them.

Isabella Bannerman’s Six Chix for the 18th of July, 2022. Essays with some mention of Six Chix are at this link.

Bill Amend’s FoxTrot for the 24th of July gets a bit more solidly mathematical, as it’s natural to think of this sort of complicated polyhedron as something mathematicians do. Geometers at least. There’s a comfortable bit of work to be done in these sorts of shapes. They sometimes have appealing properties, for instance balancing weight loads well. Building polyhedrons out of toothpicks and gumdrops, or straws and marshmallows, or some other rigid-and-soft material, is a fine enough activity. I think every mathematics department has some dusty display shelf with a couple of these.

Bill Amend’s FoxTrot for the 24th of July, 2022. This and other essays inspired by FoxTrot are at this link.

There are many shapes that Paige Fox’s construction might be. To my eye Paige Fox seems to be building a truncated icosahedron, that is to say, the soccer-ball shape. It’s an Archimedean solid, one of the family of thirteen shapes made of nonintersecting regular convex polygons. These are the shapes you discover if you go past Platonic solids. The family is named for that Archimedes, although the work in which he discussed them is now lost.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 26th of July, 2022. I have many essays describing something mentioned by Saturday Morning Breakfast Cereal at this link.

Zach Weinersmith’s Saturday Morning Breakfast Cereal for the 26th of July is a different take on the challenge of motivating students to care about mathematics. The “right approach” argument has its appeal, although it’s obviously thinking of “professor” as the only job a mathematician can hold. And even granting that none of the people who understand your work have the power to fire you, that means people who don’t understand your work do have it. Also, people shut down very hard at your promise that they could understand anything about what you do, even when you know how to express it in common-language terms. This gets old fast. I am also skeptical that women are impressed by men at bars who claim to be employed for their intellect.

Anyway, academic jobs are great and more jobs should work by their rules, which, yes, do include extremely loose set hours and built-in seasons where the amount and kind of work you do varies.

Patrick Roberts’s Todd the Dinosaur for the 5th of August, 2022. Essays with some mention of Todd the Dinosaur are at this link.

Patrick Roberts’s Todd the Dinosaur for the 5th of August is a mathematics-anxiety dream, represented with the sudden challenge to do mental arithmetic. 47 times 342 is an annoying problem, yes, but one can at least approximate it fairly well quickly. 47 is almost 50, which is half a hundred. So 47 times 342 has to be nearly half of a hundred times 342, that is, half of 34,200. This is an easy number to cut in half, though: 17,100. To get this exactly? 47 is three less than fifty, so, subtract three times 342. 342 is about a third of a thousand, we can make a better estimate by subtracting a thousand: 16,100.

If you’re really good you notice that 342 is nine more than 333, so, three times 342 is three times nine more than three times 333. That is, it’s 27 more than 999. So the 16,100 estimate is 26 more than the correct number, 16,074. And I believe if you check, you will find the card in your hand is the ace of clubs. Am I not right, professor?

And that’s a look at comic strips through to a bit over a week ago. I intend to have more soon. All my Reading the Comics posts should be at this link. Thanks for being with me for these reviews. See you again soon.

How to Add Up Powers of Numbers

Do you need to know the formula to tell you what the sum of the first N counting numbers, raised to a power? No, you do not. Not really. It can save a bit of time to know the sum of the numbers raised to the first power. Most mathematicians would know it, or be able to recreate it fast enough:

But there are similar formulas to add up, say, the counting numbers squared, or cubed, or so. And a toot on Mathstodon, the mathematics-themed instance of social network Mastodon, makes me aware of a cute paper about this. In it Dr Alessandro Mariani describes A simple mnemonic to compute sums of powers.

It’s a neat one. Mariani describes a way to use knowledge of the sum of numbers to the first power to generate a formula for the sum of squares. And then to use the sum of squares formula to generate the sum of cubes. The sum of cubes then lets you get the sub of fourth-powers. And so on. This takes a while to do if you’re interested in the sum of twentieth powers. But do you know how many times you’ll ever need to generate that formula? Anyway, as Mariani notes, this sort of thing is useful if you find yourself at a mathematics competition. Or some other event where you can’t just have the computer calculate this stuff.

Mariani’s process is a great one. Like many mnemonics it doesn’t make literal sense. It expects one to integrate and differentiate polynomials. Anyone likely to be interested in a formula for the sums of twelfth powers knows how to do those in their sleep. But they’re integrating and differentiating polynomials for which, in context, the integrals and derivatives don’t exist. Or at least don’t mean anything. That’s all right. If all you want is the right answer, it’s okay to get there by a wrong method. At least if you verify the answer is right, which the last section of Mariani’s paper does. So, give it a read if you’d like to see a neat mathematical trick to a maybe useful result.

How to Impress Someone by Multiplying Certain Big Numbers in Your Head

Mental arithmetic is fun. It has some use, yes. It’s always nice when you’re doing work to have some idea what a reasonable answer looks like. But mostly it’s fun to be able to spot, oh, 24 times 16, that’s got to be a little under 400.

I ran across this post, by Math1089, with a neat trick for certain multiplications. It’s limited in scope. Most mental-arithmetic tricks are; they have certain problems they do well and you need to remember a grab bag that covers enough to be useful. Here, the case is multiplying two numbers that start the same way, and whose ends are complements. That is, the ends add together to 10. (Or, to 100, or 1000, or some other power of two.) So, for example, you could use this trick to multiply together 41 and 49, or 64 and 66. (Or, if you needed, to multiply 2038 by 2062.)

It won’t directly solve 41 times 39, though, nor 64 times 65. But you can hack it together. 64 times 65 is 64 times 66 — you have a trick for that — minus 64. 41 times 39 is tougher, but, it’s 41 times 49 minus 41 times 10. 41 times 10 is easy to do. This is what I mean by learning a grab bag of tricks. You won’t outpace someone who has their calculator out and ready to go. But you might outpace someone who has to get their calculator out, and you’ll certainly impress them.

So it’s clever, and not hard to learn. If you feel like testing your high-school algebra prowess you can even work out why this trick works, and why it has the limits it does.

Reading the Comics, April 17, 2020: Creating Models Edition

And now let me close out a week ago, in the comics. It was a slow week and it finished on a bunch of casual mentions of mathematical topics.

Gary Larson’s The Far Side compilation “Hands Off My Bunsen Burner” features this panel creating a model of how to get rights out of wrongs. The material is a joke, but trying to find a transformation from one mathematical object to another is a reasonable enough occupation.

Gary Larson’s The Far Side, a compilation for April 2020. Essays which feature some mention of The Far Side are gathered at this link.

Ted Shearer’s Quincy rerun for the 15th is one in the lineage of strips about never using mathematics in later life. Quincy challenges us to think of a time a reporter asks the President how much is 34 times 587.

Ted Shearer’s Quincy rerun for the 15th of April 2020. It originally ran, looks like, the 19th of February, 1981. The essays where I discuss something brought up by Quincy should be at this link.

That’s an unpleasant multiplication to do. But I can figure some angles on it. 34 is just a bit over one-third of 100. 587 is just a bit under 600. So, 34 times 587 has to be tolerably near one-third of 100 times 600. So it should be something around 20,000. To get it more exact: 587 is 13 less than 600. So, 587 times one-third of a hundred will be 600 times one-third of a hundred minus 13 times one-third of a hundred. That’s one-third of 130, which is about 40. So the product has to be something close to 19,960. And the product has be some number which ends in an 8, what with 4 times 7 being 28. So the answer has to be one of 19,948, 19,958, or 19,968. And, indeed, it’s 19,958. I doubt I could do that so well during a press conference, I’ll admit. (If I wanted to be sure about that second digit, I’d have worked out: the tens unit in 34 times the ones in 587 is three times seven which is 21; the ones unit in 34 times the tens unit in 587 is four times eight which is 32; and the 4 times 7 being 28 gives me a 2 in the tens unit. So, 1 plus 2 plus 2 is 5, and there we go.)

Brian Anderson’s Dog Eat Doug for the 15th uses blackboards full of equations to represent deep thinking. I can’t make out what the symbols say. They look quite good, though, and seem to have the form of legitimate expressions.

Terri Liebenson’s The Pajama Diaries for the 17th of April, 2020. It originally ran the 20th of April, 2007. Essays inspired by something mentioned in The Pajama Diaries are at this link.

Terri Liebenson’s The Pajama Diaries for the 17th imagines creating a model for the volume of a laundry pile. The problem may seem trivial, but it reflects an important kind of work. Many processes are about how something that’s always accumulating will be handled. There’s usually a hard limit to the rate at which whatever it is gets handled. And there’s usually very little reserve, in either capacity or time. This will cause, for example, a small increase in traffic in a neighborhood to produce great jams, or how a modest rain can overflow the whole city’s sewer systems. Or how a day of missing the laundry causes there to be a week’s backlog of dirty clothes.

And a little final extra comic strip. I don’t generally mention web comics here, except for those that have fallen in with a syndicator like GoComics.com. (This is not a value judgement against web comics. It’s that I have to stop reading sometime.) But Kat Swenski’s KatRaccoon Comics recently posted this nice sequence with a cat facing her worst fear: a calculus date.

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And that’s my comics for a week ago. Later this week I’ll cover the past week’s handful of comics, in an essay at this link. Thanks for reading.

Yes, I Am Late With The Comics Posts Today

I apologize that, even though the past week was light on mathematically-themed comic strips, I didn’t have them written up by my usual Sunday posting time. It was just too busy a week, and I am still decompressing from the A to Z sequence. I’ll have them as soon as I’m able.

In the meanwhile may I share a couple of things I thought worth reading, and that have been waiting in my notes folder for the chance to highlight?

There are around 7000 people currently living in this planet who got 20 tails in a row the first time they tried flipping a coin in their life pic.twitter.com/LvUWs4jnLA

— Fermat's Library (@fermatslibrary) October 8, 2018

This Fermat’s Library tweet is one of those entertaining consequences of probability, multiplied by the large number of people in the world. If you flip twenty coins in a row there’s a one in 1,048,576 chance that all twenty will come up heads, or all twenty will come up tails. So about one in every million times you flip twenty coins, they all come up the same way. If the seven billion people in the world have flipped at least twenty coins in their lives, then something like seven thousand of them had the coins turn up heads every single one of those twenty times. That all seven billion people have tossed a coin seems like the biggest point to attack this trivia on. A lot of people are too young, or don’t have access to, coins. But there’s still going to be thousands who did start their coin-flipping lives with a remarkable streak.

Also back in October, so you see how long things have been circulating around here, John D Cook published an article about the World Series. Or any series contest. At least ones where the chance of each side winning don’t depend on the previous games in the series. If one side has a probability ‘p’ of winning any particular game, what’s the chance they’ll win a best-four-of-seven? What makes this a more challenging mathematics problem is that a best-of-seven series stops after one side’s won four games. So you can’t simply say it’s the chance of four wins. You need to account for four wins out of five games, out of six games, and out of seven games. Fortunately there’s a lot of old mathematics that explores just this.

The economist Brandford DeLong noticed the first write-up of the Prisoners Dilemma. This is one of the first bits of game theory that anyone learns, and it’s an important bit. It establishes that the logic of cooperatives games — any project where people have to work together — can have a terrible outcome. What makes the most sense for the individuals makes the least sense for the group. That a good outcome for everyone depends on trust, whether established through history or through constraints everyone’s agreed to respect.

And finally here’s part of a series about quick little divisibility tests. This is that trick where you tell what a number’s divisible by through adding or subtracting its (base ten) digits. Everyone who’d be reading this post knows about testing for divisibility by three or nine. Here’s some rules for also testing divisibility by eleven (which you might know), by seven (less likely), and thirteen. With a bit of practice, and awareness of some exceptional numbers, you can tell by sight whether a number smaller than a thousand is prime. Add a bit of flourish to your doing this and you can establish a reputation as a magical mathematician.

Reading the Comics, April 24, 2016: Mental Mathematics and Calendars Edition

Warning! I do some showing off in this installment of the Reading the Comics series. Please forgive me. I was feeling a little giddy.

Scott Hilburn’s The Argyle Sweater I had just mentioned to a friend never seems to show up in these columns anymore. And Hilburn would so reliably do strips about anthropomorphized numerals. He returns on the 20th, after a hiatus of some length I haven’t actually checked here, with a name-drop of Einstein instead. I grinned, although a good part of what amused me was the look of the guy in the lower right of the panel. Funny pictures carry a comic strip far. Formulating the theory of relativity is a tricky request. The special theory … well, to do it properly takes some sophisticated work. But it doesn’t take much beyond the Pythagorean Theorem to realize that “how long” a thing is, or a time span is, is different for different observers. That’s the most important insight, I would say, and that is easily available. General relativity, which looks at accelerations and gravity, that’s another thing. I’d be interested in a popular treatment that explained enough mathematics people could make usable estimates but that could still make sense to a lay audience. Probably it’s not possible to do this. Too bad.

Mark Tatulli’s Heart of the City just uses arithmetic because it’s a nice compact problem to give a student. It did strike me that 117 times 45 is something one could amaze people with by doing in one’s head, though. Here’s why. 117 times 100 would be easy. Multiplying by hundreds always is. 117 times 50 would be not almost as easy: that’s multiplying by 100 and dividing by two. 117 times 45 … well, that’s 117 times 50 minus 117 times 5. And if you know 117 times 50, then you know 117 times 5: it’s one-tenth that. And one-tenth of a thing is easy to find.

Therefore: 117 times 100 is 11,700. Divide that by two and that’s kind of an ugly-looking number, isn’t it? But all’s not lost. Let me use another bit of falsework: 11,700 is 12,000 minus 300. Half that is 6,000 minus 150. Therefore, half of 11,700 is 5,850. So 117 times 50 is 5,850. One-tenth of that is 585. Therefore, 117 times 45 is 5,850 minus 585. And that will be … 5,275. Ta-da!

Well, no, it isn’t. It’s 5,265. I messed up the carrying. I still think that’s doing well for multiplying ugly numbers like that without writing it down. It just won’t impress people who want the actual you know correct answer.

Mark Anderson’s Andertoons wouldn’t let me down by vanishing for a while. The 21st is not explicitly a strip about extrapolating graphs. I’ll take it as such, though. Once again the art amuses me. I like the crash-up of charted bars. Yes, I saw the Schrödinger’s Cat thing two days later.

Jef Mallett’s Frazz for the 23rd I drag into a mathematics blog because of the long historical links between calendars and mathematics. But Caulfield does talk about something that’s baffled everyone. There’s seven days to the week. There’s seven classically known heavenly bodies in the solar system, besides the Earth. Naming a day for each seems obvious now that we’ve committed to it. But why aren’t the bodies honored in order?

Geocentrism seems like, at first, a plausible reason. The ancients wouldn’t order the sky Sun-Mercury-Venus-Moon-Mars-Jupiter-Saturn. But that doesn’t help. Geocentric models of the solar system (always, so far as I’m aware) put the Moon closest, then Mercury, then Venus, the Sun, Mars, Jupiter, and Saturn.

The answer that, at least, gets repeated in histories of the calendar (for example, here, David Ewing Duncan’s The Calendar: The 5000-Year Struggle To Align The Clock And The Heavens — And What Happened To The Missing Ten Days, which was the first book I had on hand) amounts to a modular arithmetic thing. The Babylonians, if Duncan is right, named a planet-god for each hour of the day. (We treat the Moon and Sun as planets for this discussion.) The planet-gods took their hourly turn in order. If the first hour of the day is Saturn’s to rule, the next is Jupiter’s, then Mars’s, the Sun’s, Venus’s, Mercury’s, and the Moon’s. Then back to Saturn and the system keeps going like that.

So if the first hour of the day is Saturn’s, then who has the first hour of the next day? … the Sun does. If the Sun has the first hour of the day, then who has the first hour of the day after that? … the Moon. And from here you know the pattern. At least you do if you understand that English derives most of its day names from the Norse gods, matched as best they can with those of the Roman State Religion. So, Tiw matches with Mars; Woden with Mercury; Thor with Jupiter; Freya with Venus. The apparently scrambled order of days, relative to the positions of the planets, amounts to what you get if you keep adding 24 to a number by modulo 7 arithmetic.

That is, at least, the generally agreed-upon explanation. I am not aware of what actual researchers of Babylonian culture believe. Duncan, I must admit, takes a hit in his credibility by saying on the page after this that “recently chronobiologists have discovered that the seven-day cycle … may also have biological precedents”. I’m sorry but I just don’t believe him, or whoever he got that from.

Kevin Fagan’s Drabble for the 24th amuses me by illustrating the common phenomenon. We have all taken out the calculator (or computer) to do some calculation that really doesn’t need it. I understand and am sympathetic. It’s so obviously useful to let the calculator work out 117 times 45 and get it right instantly. It’s easy to forget sometimes it’s faster to not bother with the calculator. We are all of us a little ridiculous.

Reading the Comics, August 10, 2015: How People Think Edition

Today’s installment of Reading the Comics has a bunch of strips that seem to touch on human psychology. That properly could always be said; what we know of mathematics is what humans have thought about. But sometimes the link between a mathematical topic and human psychology is more obvious.

Wes Molebash’s Molebashed (August 5) is a reminder that one can find interesting mental arithmetic problems anywhere. This does not mean they’re always welcome. But they can still be fun to do. For example while walking through a parking lot I noticed another state’s license plate and wondered how many six-letter combinations you could get. Well, that’s 26⁶, obviously, but how big a number is that? Working out that sort of thing is why people have to repeat what they’re saying to me.

Mark Pett’s Mr Lowe (August 6, rerun from sometime in 2000) has a student complaining the mathematics books are two years old. The complaint is absurd but also kind of sensible. Mathematical truths are immortal, or at least they are once they’re proven. Whether something is proven is, to an extent, a cultural construct: it takes an incredible load of work to actually prove something rigorously with every step in place. We usually are content if we show enough reasoning to be confident that every step could be filled in if need be. More a matter of taste, though, is whether these truths are interesting. As an example, I mentioned just a few posts ago the versine function. There are computations which, if you’re doing them by hand, are best done with the versine function or a table of values of the versine function. But we don’t need to do that sort of work anymore, and the versine function has plunged into obscurity. Nothing that we knew about versines has stopped being true. But we’d be eccentric, at least, to make it a part of a trigonometry course in the way someone 150 years ago might have. Mathematics is not culturally neutral. Few interesting things are.

Kieran Meehan’s Pros and Cons for the 7th of August, 2015.

Kieran Meehan’s Pros and Cons (August 7) is a probability joke. As often happens, the probability joke is built on the gambler’s fallacy. The fallacy in this case is the supposition that if one hasn’t had an accident in an unusually long while, then one must be due. Properly, though, we should ask whether accidents are independent events. If they are independent — if the chance of having an accident does not change based on whether you had an accident yesterday, or in the past week, or in the past year, or so on — then it’s silly to say you’re “due” for one. If your rate of accidents is lower than expected, you’re just having a lucky streak is all. However, I can imagine the chance of having an accident not being independent. I can imagine going a long time without accidents making someone careless about normal risks, or inexperienced in judging new ones, and that might make an accident more likely that one expects. It’s difficult to answer a probability question without understanding human psychology.

John Graziano’s Ripley’s Believe It or Not (August 7) claims there are over 26,000 possible outcomes of tic-tac-toe. I think the claim is poorly worded, though. If by an “outcome” of a tic-tac-toe game we mean the arrangement of X and O marks then there are at most 19,683 outcomes — each of the nine cells contains an X, an O, or is left blank. That’s an overestimate, though. A grid of nine X’s can’t be a legal outcome of a game, after all; nor can one that has two X’s, one O, and six blank spaces. There have to be at least three X’s and at least two O’s, and at most four blank spaces. The number of X’s can be equal to or one greater than the number of O’s. This removes a lot of possibilities.

I think what Graziano’s Ripley’s wants to claim is there are over 26,000 different tic-tac-toe games. This I can more readily believe. There are 9 possible spaces the first player can take on the first turn; there are 8 choices for the second player on the first turn. There are 7 choices for the first player on the second turn; there are 6 choices for the second player on the second turn. And so on. So there are at most 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 possible ways to play out the game; that’s a total of 362,880 possibilities. But not all those possibilities are needed. If a game’s won after two and a half turns, it stops, and a lot of possible continuations are voided. I don’t have a good estimate of how many those are. And we might choose to rule out symmetries. The game in which X fills out the top row while O tries the center isn’t really different to the game in which X fills out the bottom row while O takes the center. For that matter, it’s not different to the one where X fills in the right column while O fills in the center column. If you don’t count symmetries like this as different games, then we have fewer games altogether. So if that is what Graziano means, then 26,000 may be a fair estimate of tic-tac-toe games.

That, by the way, is the strip that gave me the most to think about of this set.

Rick Kirkman and Jerry Scott’s Baby Blues for the 8th of August, 2015.

Rick Kirkman and Jerry Scott’s Baby Blues (August 8) is another installment of Kids Doing Mathematics During Summer Vacation. This is almost the theme of the summer in mathematics comics. Possibly it’s the theme of every summer.

Bill Amend’s FoxTrot (August 9) is one of those odd jokes that also is a pretty good business opportunity. Jason Fox proposes some of the many shapes that could, in principle, hold ice cream. I believe hemispheres at least are available, actually, at least to restaurants. But some of these shapes, such as pyramids or dodecahedrons or such, seem like they could be made and just happen not to have been. (Well, half-dodecahedrons, anyway.) That probably reflects that a cone or similarly narrow-based shape forces more of a given amount of ice cream to overflow the top of the cone, suggesting abundance. Geometric possibilities must give way to making the product look bigger.

Reading the Comics, September 15, 2014: Are You Trying To Overload Me Edition

One of the little challenges in writing about mathematics-themed comics is one of pacing: how often should I do a roundup? Posting weekly, say, helps figure out a reasonable posting schedule for those rare moments when I’m working ahead of deadline, but that leaves the problem of weeks that just don’t have anything. Waiting for a certain number of comics before writing about them seems more reasonable, but then I have to figure how many comics are enough. I’ve settled into five-to-six as my threshold for a new post, but that can mean I have weeks where it seems like I’m doing nothing but comic strips posts. And then there’s conditions like this one where Comic Strip Master Command had its cartoonists put up just enough that I’d started composing a fresh post, and then tossed in a whole bunch more the next day. It’s like they’re trying to shake me by having too many strips to write about. I’d have though they’d be flattered to have me writing about them so.

Bud Blake’s _Tiger_ for the 11th of September, 2014.

Bud Blake’s Tiger (September 11, rerun) mentions Tiger as studying the times tables and points out the difference between studying a thing and learning it.

Marc Anderson’s Andertoons (September 12) belongs to that vein of humor about using technology words to explain stuff to kids. I admit I’m vague enough on the concept of mashups that I can accept that it might be a way of explaining addition, but it feels like it might also be a way of describing multiplication or for that matter the composition of functions. I suppose the kids would be drawn as older in those cases, though.

Bill Amend’s FoxTrot (September 13, rerun) does a word problem joke, but it does have the nice beat in the penultimate panel of Paige running a sanity check and telling at a glance that “two dollars” can’t possibly be the right answer. Sanity checks are nice things to have; they don’t guarantee against making mistakes, but they at least provide some protection against the easiest mistakes, and having some idea of what an answer could plausibly be might help in working out the answer. For example, if Paige had absolutely no idea how to set up equations for this problem, she could reason that the apple and the orange have to cost something from 1 to 29 cents, and could try out prices until finding something that satisfies both requirements. This is an exhausting method, but it would eventually work, too, and sometimes “working eventually” is better than “working cleverly”.

Bill Schorr’s The Grizzwells (September 13) starts out by playing on the fact that “yard” has multiple meanings; it also circles around one of those things that distinguishes word problems from normal mathematics. A word problem, by convention, normally contains exactly the information needed to solve what’s being asked — there’s neither useless information included nor necessary information omitted, except if the question-writer has made a mistake. In a real world application, figuring out what you need, and what you don’t need, is part of the work, possibly the most important part of the work. So to answer how many feet are in a yard, Gunther (the bear) is right to ask more questions about how big the yard is, as a start.

Steve Kelly and Jeff Parker’s _Dustin_ for the 14th of September, 2014.

Steve Kelley and Jeff Parker’s Dustin (September 14) is about one of the applications for mental arithmetic that people find awfully practical: counting the number of food calories that you eat. Ed’s point about it being convenient to have food servings be nice round numbers, as they’re easier to work with, is a pretty good one, and it’s already kind of accounted for in food labelling: it’s permitted (in the United States) to round off calorie counts to the nearest ten or so, on the rather sure grounds that if you are counting calories you’d rather add 70 to the daily total than 68 or 73. Don’t read the comments thread, which includes the usual whining about the Common Core and the wild idea that mental arithmetic might be well done by working out a calculation that’s close to the one you want but easier to do and then refining it to get the accuracy you need.

Mac and Bill King’s Magic In A Minute kids activity panel (September 14) presents a magic trick that depends on a bit of mental arithmetic. It’s a nice stunt, although it is certainly going to require kids to practice things because, besides dividing numbers by 4, it also requires adding 6, and that’s an annoying number to deal with. There’s also a nice little high school algebra problem to be done in explaining why the trick works.

Bill Watterson’s Calvin and Hobbes (September 15, rerun) includes one of Hobbes’s brilliant explanations of how arithmetic works, and if I haven’t wasted the time spent memorizing the strips where Calvin tries to do arithmetic homework then Hobbes follows up tomorrow with imaginary numbers. Can’t wait.

Jef Mallet’s Frazz (September 15) expresses skepticism about a projection being made for the year 2040. Extrapolations and interpolations are a big part of numerical mathematics and there’s fair grounds to be skeptical: even having a model of whatever your phenomenon is that accurately matches past data isn’t a guarantee that there isn’t some important factor that’s been trivial so far but will become important and will make the reality very different from the calculations. But that hardly makes extrapolations useless: for one, the fact that there might be something unknown which becomes important is hardly a guarantee that there is. If the modelling is good and the reasoning sound, what else are you supposed to use for a plan? And of course you should watch for evidence that the model and the reality aren’t too very different as time goes on.

Gary Wise and Lance Aldrich’s Real Life Adventures (September 15) describes mathematics as “insufferable and enigmatic”, which is a shame, as mathematics hasn’t said anything nasty about them, now has it?

Reading the Comics, June 22, 2014: Name-Dropping Stuff Edition

Comic Strip Master Command apparently really is ordering strips to finish their mathematics jokes before the summer vacation sets in, based on how many we’ve gotten in the past week. I confess this set doesn’t give me so much to write about; it’s more a set of mathematics things getting name-dropped. And there’s always something, isn’t there?

Tom Thaves’s Frank and Ernest (June 17) showcases a particularly severe form of math anxiety. I’m sympathetic to people who’re afraid of mathematics, naturally; it’s rotten being denied a big and wonderful and beautiful part of human ingenuity. I don’t know where math anxiety comes from, although I’d imagine a lot of it comes from that mix of doing something you aren’t quite sure you’re doing correctly and being hit too severely with a sense of rejection in the case that you did it wrong. I’d like to think that recreational mathematics puzzles would help overcome that, but I have no evidence that it does, just my hunch that getting to play with numbers and pictures and logic puzzles is good for you.

Russell Myers’ Broom Hilda (June 18) taunts the schoolkid Nerwin with the way we “used to do math with our brains instead of calculators”. One hesitates to know too much about the continuity of Broom Hilda, but I believe she’s over a thousand years old and so when she was Nerwin’s age they didn’t even have Arabic numerals just yet. I’ll assume there’s some way she’d have been in school then. (Also, given how long Broom Hilda‘s been running Nerwin did used to be in classes that did mathematics without calculators.)

Chris Browne’s Hagar the Horrible for the 19th of June, 2014.

Chris Brown’s Hagar the Horrible (June 19) tries to get itself cut out and put up on the walls of math tutors’ offices. Good luck.

Tom Batiuk and Chuck Ayers’s Crankshaft for the 20th of June, 2014.

Tom Batiuk and Chuck Ayers’ Crankshaft (June 20) spent a couple days this week explaining how he just counts on fingers to do his arithmetic. It’s a curious echo of the storyline several years ago revealing Crankshaft suffered from Backstory Illiteracy, in which we suddenly learned he had gone all his life without knowing how to read. I hesitate to agree with him but, yeah, there’s no shame in counting on your fingers if that does all the mathematics you need to do and you get the answers you want reliably. I don’t know what his long division thing is; if it weren’t for Tom Batiuk writing the comic strip I’d call it whimsy.

Keith Knight’s The Knight Life carried on with the story of the personal statistician this week. I think the entry from the 20th is most representative. It’s fine, and fun, to gather all kinds of data about whatever you encounter, but if you aren’t going to study the data and then act on its advice you’re wasting your time. The personal statistician ends up quitting the job.

Steve McGarry’s kid-activity feature KidTown (June 22) promotes the idea of numbers as a thing to notice in the newspapers, and includes a couple of activities, one featuring a maze to be navigated by way of multiples of seven. It also has one of those math tricks where you let someone else pick a number, give him a set of mathematical operations to do, and then you can tell them what the result is. It seems to me working out why that scheme works is a good bit of practice for someone learning algebra, and developing your own mathematics trick that works along this line is further good practice.

Listening To Vermilion Sands

My Beloved is reading J G Ballard’s Vermillion Sands; early in one of the book’s stories is a character wondering if an odd sound comes from one of the musical … let’s call it instruments, one with a 24-octave range. We both thought, wow, that’s a lot of range. Is it a range any instrument could have?

As we weren’t near our computers this turned into a mental arithmetic problem. It’s solvable in principle because, if you know the frequency of one note, then you know the frequency of its counterpart one octave higher (it’s double that), and one octave lower (it’s half that). It’s not solvable, at this point, because we don’t have any information about what the range is supposed to be. So here’s roughly how we worked it out.

The note A above middle C is 440 Hertz, or at least you can use that for tuning ever since the International Standards Organization set that as a tuning standard in 1953. (As with any basically arbitrary standard this particular choice is debatable, although, goodness but this page advocating a 432 Hertz standard for A doesn’t do itself any favors by noting that “440 Hz is the unnatural standard tuning frequency, removed from the symmetry of sacred vibrations and overtones that has declared war on the subconscious mind of Western Man” and, yes, Nikola Tesla and Joseph Goebbels turn up in the article because you might otherwise imagine taking it seriously.) Anyway, it doesn’t matter; 440 is just convenient as it’s a number definitely in hearing range.

So I’m adding the assumption that 440 Hz is probably in the instrument’s range. And I’ll work on the assumption that it’s right in the middle of the range, that is, that we should be able to go down twelve octaves and up twelve octaves, and see if that assumption leads me to any problems. And now I’ve got the problem defined well enough to answer: is 440 divided by two to the twelfth power in human hearing range? Is 440 times two to the twelfth power in range?

I’m not dividing 440 by two a dozen times; I might manage that with pencil and paper but not in my head. But I also don’t need to. Two raised to the tenth power is pretty close to 1,000, as anyone who’s noticed that the common logarithm of two is 0.3 could work out. Remembering a couple approximations like that are key to doing any kind of real mental arithmetic; it’s all about turning the problem you’re interested in into one you can do without writing it down.

Another key to this sort of mental arithmetic is noticing that two to the 12th power is equal to two to the second power (that is, four) times two to the tenth power (approximately 1,000). In algebra class this was fed to you as something like “a^{x + y} = (a^x)(a ^y)”, and it’s the trick that makes logarithms a concept that works.

Getting back to the question, 440 divided by two twelve times over is going to be about 440 divided by 4,000, which is going to be close enough to one-tenth Hertz. There’s no point working it out to any more exact answer, since this is definitely below the range of human hearing; I think the lower bound is usually around ten to thirty Hertz.

Well, no matter; maybe the range of the instrument starts higher up and keeps on going. To see if there’s any room, what’s the frequency of a note twelve octaves above the 440-Hertz A?

That’s going to be 440 Hertz times 4,000, which to make it simpler I’ll say is something more than 400 times 4000. The four times four is easy, and there’s five zeroes in there, so, that suggests an upper range on the high side of 1,600,000 Hertz. Again, I’m not positive the upper limit of human hearing but I’m confident it’s not more than about 30,000 Hertz, and I leave space below for people who know what it is exactly to say. There’s just no fitting 24 octaves into the human hearing range.

So! Was Ballard just putting stuff into his science fiction story without checking whether the numbers make that plausible, if you can imagine a science fiction author doing such a thing?

It’s conceivable. It’s also possible Ballard was trying to establish the character was a pretentious audiophile snob who imagines himself capable of hearing things that no, in fact, can’t be discerned. However, based on the setting … the instruments producing music in this story (and other stories in the book), set in the far future, include singing plants and musical arachnids and other things that indicate not just technology but biology has changed rather considerably. If it’s possible to engineer a lobster that can sing over a 24 octave range, it’s presumably possible to engineer a person who can listen to it.

How Big Is This Number? Answered

My little question about just how big a number was got answered just exactly right by John Friedrich, so if you wondered about how I could say a number took about seven million digits just to write out, there’s your answer. Friedrich gives it as a number with 6,846,169 digits, and I agree. Better, the calculator I found which was able to handle this (MatCalcLite, a free calculator app I have on my iPad) agrees too: it claims that is about which has that magic 6,846,169 digits.

Friedrich uses logarithms to work it out, and this is one of the things logarithms are good for in these days when you don’t generally need them to do multiplications and divisions. You can look at logarithms as letting you evaluate the lengths of numbers — how many digits they need to work out — rather than the numbers themselves, and this brings to the field of accessibility numbers that would otherwise be too big to work with, even on the calculator. (Another thing logarithms are good for is that they’re quite nice to work with if you have to do calculus, so once you’re comfortable with them, you start looking for chances to slip them into analysis.)

One nagging little point about Friedrich’s work, though, is that you need to know the logarithm of 3 to work it out. (Also you need the logarithm of 10, or you could try using the common logarithm — the logarithm base ten — of 3 instead.) For finding the actual number that’s fine; trying to get this answer with any precision without looking up the logarithm of 3 is quirky if not crazy.

But what if you want to do this purely by the joys of mental arithmetic? Could you work out without finding a table of logarithms? Obviously you can’t if you want a really precise answer, and here counts as precise, but could you at least get a good idea of how big a number it is?

Reading the Comics, 16 May 2013

It’s a good time for another round of comic strip reading, particularly I haven’t had the time to think in detail about all the news in number theory that’s come out this past week, and that I’m not sure whether I should go into explaining arc lengths after I trapped at least one friend into trying to work out the circumference of an ellipse (you can’t do it either, but there are a lot of curves you could). I also notice I’m approaching that precious 10,000th blog hit here, so I can get back to work verifying that law about random data starting with the digit 1.

Berkeley Breathed’s Bloom County (May 2, rerun) throws up a bunch of mathematical symbols with the intention of producing a baffling result, so that Milo can make a clean getaway from Freida. The splendid thing to me, though, is that Milo’s answer — “log 10 times 10 to the derivative of 10,000” — actually does parse, if you read it a bit charitably. The “log 10” bit we can safely suppose to mean the logarithm base 10, because the strip originally ran in 1981 or so when there was still some use for the common logarithm. These days, we have calculators, and “log” is moving over to be the “natural logarithm”, base e, what was formerly denoted as “ln”.

Why Someone Should Take That Deal

Let me start answering my Deal or No Deal-based question by just pointing to Chiaroscuro’s answer, which does the arithmetic exactly right and comes to a quite sensible conclusion from it. This leaves me feeling like I’m not quite earning my pay here, so let me go into further depth and ask that someone pay me.

How To Multiply By 365 In Your Head

Kevin Fagin’s Drabble from Sunday poses a nice bit of recreational mathematics, the sort of thing one might do for amusement: Ralph Drabble tries to figure how long he’s spent waiting at one traffic light. I want to talk about some of the mental arithmetic tricks I’d use to get through the puzzle without missing the light’s change. In the spirit of the thing I’m doing the calculations for this only in my head, though I admit checking with a calculator afterward to see if I got close.