I didn’t make noise about it, but last Sunday’s mathematics comic strip roundup was short one day. I was away from home and normal computer stuff Saturday. So I posted without that day’s strips under review. There was just the one, anyway.
Also I want to remind folks I’m doing another Mathematics A To Z, and taking requests for words to explain. There are many appealing letters still unclaimed, including ‘A’, ‘T’, and ‘O’. Please put requests in over on that page because. It’s easier for me to keep track of what’s been claimed that way.
Matt Janz’s Out of the Gene Pool rerun for the 15th missed last week’s cut. It does mention the Law of Cosines, which is what the Pythagorean Theorem looks like if you don’t have a right triangle. You still have to have a triangle. Bobby-Sue recites the formula correctly, if you know the notation. The formula’s . Here ‘a’ and ‘b’ and ‘c’ are the lengths of legs of the triangle. ‘C’, the capital letter, is the size of the angle opposite the leg with length ‘c’. That’s a common notation. ‘A’ would be the size of the angle opposite the leg with length ‘a’. ‘B’ is the size of the angle opposite the leg with length ‘b’. The Law of Cosines is a generalization of the Pythagorean Theorem. It’s a result that tells us something like the original theorem but for cases the original theorem can’t cover. And if it happens to be a right triangle the Law of Cosines gives us back the original Pythagorean Theorem. In a right triangle C is the size of a right angle, and the cosine of that is 0.
That said Bobby-Sue is being fussy about the drawings. No geometrical drawing is ever perfectly right. The universe isn’t precise enough to let us draw a right triangle. Come to it we can’t even draw a triangle, not really. We’re meant to use these drawings to help us imagine the true, Platonic ideal, figure. We don’t always get there. Mock proofs, the kind of geometric puzzle showing something we know to be nonsense, rely on that. Give chalkboard art a break.
Samson’s Dark Side of the Horse for the 17th is the return of Horace-counting-sheep jokes. So we get a π joke. I’m amused, although I couldn’t sleep trying to remember digits of π out quite that far. I do better working out Collatz sequences.
Hilary Price’s Rhymes With Orange for the 19th at least shows the attempt to relieve mathematics anxiety. I’m sympathetic. It does seem like there should be ways to relieve this (or any other) anxiety, but finding which ones work, and which ones work best, is partly a mathematical problem. As often happens with Price’s comics I’m particularly tickled by the gag in the title panel.
Norm Feuti’s Gil rerun for the 19th builds on the idea calculators are inherently cheating on arithmetic homework. I’m sympathetic to both sides here. If Gil just wants to know that his answers are right there’s not much reason not to use a calculator. But if Gil wants to know that he followed the right process then the calculator’s useless. By the right process I mean, well, the work to be done. Did he start out trying to calculate the right thing? Did he pick an appropriate process? Did he carry out all the steps in that process correctly? If he made mistakes on any of those he probably didn’t get to the right answer, but it’s not impossible that he would. Sometimes multiple errors conspire and cancel one another out. That may not hurt you with any one answer, but it does mean you aren’t doing the problem right and a future problem might not be so lucky.
Zach Weinersmith’s Saturday Morning Breakfast Cereal rerun for the 19th has God crashing a mathematics course to proclaim there’s a largest number. We can suppose there is such a thing. That’s how arithmetic modulo a number is done, for one. It can produce weird results in which stuff we just naturally rely on doesn’t work anymore. For example, in ordinary arithmetic we know that if one number times another equals zero, then either the first number or the second, or both, were zero. We use this in solving polynomials all the time. But in arithmetic modulo 8 (say), 4 times 2 is equal to 0.
And if we recklessly talk about “infinity” as a number then we get outright crazy results, some of them teased in Weinersmith’s comic. “Infinity plus one”, for example, is “infinity”. So is “infinity minus one”. If we do it right, “infinity minus infinity” is “infinity”, or maybe zero, or really any number you want. We can avoid these logical disasters — so far, anyway — by being careful. We have to understand that “infinity” is not a number, though we can use numbers growing infinitely large.
Induction, meanwhile, is a great, powerful, yet baffling form of proof. When it solves a problem it solves it beautifully. And easily, too, usually by doing something like testing two special cases. Maybe three. At least a couple special cases of whatever you want to know. But picking the cases, and setting them up so that the proof is valid, is not easy. There’s logical pitfalls and it is so hard to learn how to avoid them.
Jon Rosenberg’s Scenes from a Multiverse for the 19th plays on a wonderful paradox of randomness. Randomness is … well, unpredictable. If I tried to sell you a sequence of random numbers and they were ‘1, 2, 3, 4, 5, 6, 7’ you’d be suspicious at least. And yet, perfect randomness will sometimes produce patterns. If there were no little patches of order we’d have reason to suspect the randomness was faked. There is no reason that a message like “this monkey evolved naturally” couldn’t be encoded into a genome by chance. It may just be so unlikely we don’t buy it. The longer the patch of order the less likely it is. And yet, incredibly unlikely things do happen. The study of impossibly unlikely events is a good way to quickly break your brain, in case you need one.