## 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.

## How Long Is A Bad Ellipse Question?

Something came to mind while thinking about that failed grading scheme for multivariable calculus. I’d taught it two summers, and the first time around — when I didn’t try the alternate grading scheme — I made what everyone assured me was a common mistake.

One of the techniques taught in multivariable calculus is how to compute the length of a curve. There are a couple of ways of doing this, but you can think of them as variations on the same idea: imagine the curve as a track, and imagine that there’s a dot which moves along that track over some stretch of time. Then, if you know how quickly the dot is moving at each moment in time, you can figure out how long the track is, in much the same way you’d know that your parents’ place is 35 miles away if it takes you 35 minutes of travelling at 60 miles per hour to get there. There are details to be filled in here, which is why this is fit in an advanced calculus course.

Anyway, the introduction of this, and the homeworks, start out with pretty simple curves — straight lines, for example, or circles — because they’re easy to understand, and the student can tell offhand if the answer she got was right, and the calculus involved is easy. You can focus energy on learning the concept instead of integrating bizarre or unpleasant functions. But this also makes it harder to come up with a fresh problem for the exams: the student knowing how to find the length of a parabola segment or the circumference of a circle might reflect mastering the idea, or just that they remembered it from class.

So for the exam I assigned a simple variant, something we hadn’t done in class but was surely close enough that I didn’t need to work the problem out before printing up and handing out the exams. I’m sure it will shock you that an instructor might give out on an exam a problem he hasn’t actually solved already, but, I promise you, sometimes even teachers who aren’t grad students taking summer courses will do this. Usually it’s all right. Here’s where it wasn’t.