I confess I’m dissatisfied with this batch of Reading the Comics posts. I like having something like six to eight comics for one of these roundups. But there was this small flood of mathematically-themed comics on the 6th of December. I could either make do with a slightly short edition, or have an overstuffed edition. I suppose it’s possible to split one day’s comics across two Reading the Comics posts, but that’s crazy talk. So, a short edition today.
Jef Mallett’s Frazz for the 4th of December was part of a series in which Caulfield resists learning about reciprocals. The 4th offers a fair example of the story. At heart the joke is just the student-resisting-class, or student-resisting-story-problems. It certainly reflects a lack of motivation to learn what they are.
We use reciprocals most often to write division problems as multiplication. “a ÷ b” is the same as “a times the reciprocal of b”. But where do we get the reciprocal of b from? … Well, we can say it’s the multiplicative inverse of b. That is, it’s whatever number you have to multiply ‘b’ by in order to get ‘1’. But we’re almost surely going to find that taking 1 and dividing it by b. So we’ve swapped out one division problem for a slightly different one. This doesn’t seem to be getting us anywhere.
But we have gotten a new idea. If we can define the multiplication of things, we might be able to get division for almost free. Could we divide one matrix by another? We can certainly multiply a matrix by the inverse of another. (There are complications at work here. We’ll save them for another time.) A lot of sets allow us to define things that make sense as addition and multiplication. And if we can define a complicated operation in terms of addition and multiplication … If we follow this path, we get to do things like define the cosine of a matrix. Then we just have to figure out why we’d want have a cosine of a matrix.
There’s a simpler practical use of reciprocals. This relates to numerical mathematics, computer work. Computer chips do addition (and subtraction) really fast. They do multiplication a little slower. They do division a lot slower. Division is harder than multiplication, as anyone who’s done both knows. However, dividing by (say) 4 is the same thing as multiplying by 0.25. So if you know you need to divide by a number a lot, then it might make for a faster program to change division into multiplication by a reciprocal. You have to work out the reciprocal, but if you only have to do that once instead of many times over, this might make for faster code. Reciprocals are one of the tools we can use to change a mathematical process into something faster.
(In practice, you should never do this. You have a compiler that does this, and you should let it do its work. But it’s enlightening to know these are the sorts of things your compiler is looking for when it turns your code into something the computer does. And looking for ways to do the same work in less time is a noble side of mathematics.)
Charles Schulz’s Peanuts for the 4th of December (originally from 1968, on the same day) sees Peppermint Patty’s education crash against a word problem. It’s another problem in motivating a student to do a word problem. I admit when I was a kid I’d have been enchanted by this puzzle. But I was a weird one.
Dave Coverly’s Speed Bump for the 4th of December is a mathematics-symbols joke as applied to toast. I think you could probably actually sell those. At least the greater-than and the less-than signs. The approximately-equal-to signs would be hard to use. And people would think they were for bacon anyway.
Ruben Bolling’s Super-Fun-Pak Comix for the 4th of December showcases Young Albert Einstein. That counts as mathematical content, doesn’t it? The strip does make me wonder if they’re still selling music CDs and other stuff for infant or even prenatal development. I’m skeptical that they ever did any good, but it isn’t a field I’ve studied.
Bill Whitehead’s Free Range for the 5th of December uses a blackboard full of mathematical and semi-mathematical symbols to denote “stuff too complicated to understand”. The symbols don’t parse as anything. It is authentic to mathematical work to sometimes skip writing all the details of a thing and write in instead a few words describing it. Or to put in an abbreviation for the thing. That often gets circled or boxed or in some way marked off. That keeps us from later on mistaking, say, “MUB” as the product of M and U and B, whatever that would mean. Then we just have to remember we meant “minimum upper bound” by that.