Reading the Comics, December 5, 2014: Good Questions Edition
This week’s bundle of mathematics-themed comic strips has a pretty nice blend, to my tastes: about half of them get at good and juicy topics, and about half are pretty quick and easy things to describe. So, my thanks to Comic Strip Master Command for the distribution.
Bill Watterson’s Calvin and Hobbes (December 1, rerun) slips in a pretty good probability question, although the good part is figuring out how to word it: what are the chances Calvin’s Dad was thinking of 92,376,051 of all the possible numbers out there? Given that there’s infinitely many possible choices, if every one of them is equally likely to be drawn, then the chance he was thinking of that particular number is zero. But Calvin’s Dad couldn’t be picking from every possible number; all humanity, working for its entire existence, will only ever think of finitely many numbers, which is the kind of fact that humbles me when I stare too hard at it. And people, asked to pick a number, have ones they prefer: 37, for example, or 17. Christopher Miller’s American Cornball: A Laffopedic Guide To The Formerly Funny (a fine guide to jokes that you see lingering around long after they were actually funny) notes that what number people tend to pick seems to vary in time, and in the early 20th century 23 was just as funny a number as you could think of on a moment’s notice.
Zach Weinersmith’s Saturday Morning Breakfast Cereal (December 1) is entitled “how introductory physics problems are written”, and yeah, that’s about the way that a fair enough problem gets rewritten so as to technically qualify as a word problem. I think I’ve mentioned my favorite example of quietly out-of-touch word problems, a 1970s-era probability book which asked the probability of two out of five transistors in a radio failing. That was blatantly a rewrite of a problem about a five-vacuum-tube radio (my understanding is many radios in that era used five tubes) and each would have a non-negligible chance of failing on any given day. But that’s a slightly different problem, as the original question would have made good sense when it was composed, and it only in the updating became ridiculous.
Julie Larson’s The Dinette Set (December 2) illustrates one of the classic sampling difficulties: how can something be generally true if, in your experience, it isn’t? If you make the reasonable assumption that there’s nothing exceptional about you, then, shouldn’t your experience of, say, fraction of people who exercise, or average length of commute, or neighborhood crime rate be tolerably close to what’s really going on? You could probably build an entire philosophy-of-mathematics course around this panel before even starting the question of how do you get a fair survey of a population.
Scott Hilburn’s The Argyle Sweater (December 3) tells a Roman numeral joke that actually I don’t remember encountering before. Huh.
Samson’s Dark Side Of The Horse (December 3) does some play with mathematical symbols and of course I got distracted by thinking what kind of problem Horace was working on in the first panel; it looks obvious to me that it’s something about the orbit of one body around another. In principle, it might be anything, since the great discovery of algebra is that you can replace numbers with symbols like “a” and work out relations without having to know anything about them. “G”, for example, tends to mean the gravitational constant of the universe, and “GM” makes this identification almost certain: gravitation problems need the masses of a main body, like a planet, and a smaller body, like a satellite, and that’s usually represented as either m1 and m2 or as M and m.
In orbital mechanics problems, “a” often refers to the semimajor axis — the long diameter of the ellipse the orbiting body makes — and “e” the eccentricity — a measure of how close to a circle the ellipse is (an eccentricity of zero means it’s a circle — but the fact that there’s subscripts of k makes that identification suspect: subscripts are often used to distinguish which of multiple similar things you mean to talk about, and if it’s just one body orbiting the other there’s no need for that. So what is Horace working on?
The answer is: Horace is working on an orbital perturbation problem, describing how far from the true circular orbit a satellite will drift when you consider things like atmospheric drag and the slightly non-spherical shape of the Earth. ak is still a semi-major axis and ek the eccentricity, but of the little changes from the original orbit, rather than the original orbit themselves. And now I wonder if Samson plucked the original symbol just because it looked so graphically pleasant, or if Samson was slipping in a further joke about the way an attractive body will alter another body’s course.
Jenny Campbell’s Flo and Friends (December 4) offers a less exciting joke: it’s a simple word problem joke, playing on the ambiguity of “calculate how many seconds there are in the year”. Now, the dull way to work this out is to multiply 60 seconds per minute times 60 minutes per hour times 24 hours per day times 365 (or 365.25, or 365.2422 if you want to start that argument) days per year. But we can do almost as well and purely in our heads, if we remember that a million seconds is almost twelve days long. How many twelve-day stretches are there in a year? Well, right about 31 — after all, the year is (nearly) 12 groups of 31 days, and therefore it’s also 31 groups of 12 days. Therefore the year is about 31 million seconds long. If we pull out the calculator we find that a 365-day year is 31,536,000 seconds, but isn’t it more satisfying to say “about 31 million seconds” like we just did?
John Deering’s Strange Brew (December 4) took me the longest time to work out what the joke was supposed to be. I’m still not positive but I think it’s just one colleague sneering at the higher mathematics of another.
Patrick Roberts’s Todd the Dinosaur (December 5) discovers that some numbers are quite big ones, actually. There is a challenge in working with really big numbers, even if they’re usually bigger than 2. Usually we’re not interested in a number by itself, and would rather do some kind of calculation with it, and that’s boring to do too much of, but a computer can only work with so many digits at once. The average computer uses floating point arithmetic schemes which will track, at most, about 19 decimal digits, on the reasonable guess that twenty decimal digits is the difference between 3.1415926535897932384 and 3.1415926535897932385 and how often is that difference — a millionth of a millionth of a millionth of a percent — going to matter? If it does, then, you do the kind of work that gets numerical mathematicians their big paydays: using schemes that work with more digits, or chopping up a problem so that you never have to use all 86 relevant digits at once, or rewriting your calculation so that you don’t need so many digits of accuracy all at once.
Daniel Beyer’s Offbeat Comics (December 5) gives a couple of ways to express the number 4 — including, look closely, holding up fingers — as part of a joke about the driver being a former mathematics teacher.
Greg Cravens’s The Buckets (December 5) is the old, old, old joke about never using algebra in real life. Do English teachers get this same gag about never using the knowledge of how to diagram sentences? In any case, I did use my knowledge of sentence-diagramming, and for the best possible application: I made fun of a guy on the Internet with it.
I advise against reading the comments — I mean, that’s normally good advice, but comic strips attract readers who want to complain about how stupid kids are anymore and strips that mention education give plenty of grounds for it — but I noticed one of the early comments said “try to do any repair at home without some understanding of it”. I like the claim, but, I can’t think of any home repair I’ve done that’s needed algebra. The most I’ve needed has been working out the area of a piece of plywood I needed, but if multiplying length by width is algebra than we’ve badly debased the term. Even my really ambitious project, building a PVC-frame pond cover, isn’t going to be one that uses algebra unless we take an extremely generous view of the subject.