Burn Math Class: And Reinvent Mathematics for Yourself (2016)

4. On Circles and Giving Up

4.1. A Conceptual Centrifuge

4.1.1Sometimes “Solving” Is Really Just Giving Up

A centrifuge is a really neat machine. If you feed it a fluid that contains a bunch of different stuff that has been mixed together, then the centrifuge can separate out the parts by spinning the fluid very fast. Much of mathematics as it is typically presented in high school and university courses is like that mixed-up fluid. It contains a cloudy, unappetizing mixture of beautiful necessary truths, on the one hand, and unnecessary historical accidents, on the other. What we need is a conceptual centrifuge: a way of separating out the timeless relationships between ideas, from the things that might turn out very differently if we rewound the clock, stirred up the universe, and let human history play its course out again. Let’s start with an example to illustrate what I mean.

Figure 4.1: This problem seems really boring. But wait! It’s a conceptual centrifuge.

Say we’ve got a circle inside a square (see Figure 4.1). How much of the square is taken up by the circle? We’re in that little room in our heads where there’s no math except what we’ve invented ourselves. We can’t just quote something that someone told us if we can’t think of how to invent it from the ground up. So what does it even mean to ask “How much of the square is taken up by the circle?” Are we asking for the area of the square minus the area of the circle? Maybe. That’s one way to answer it. We could just list the difference, and say “All but this much is taken up by the circle.” But notice that if we answer the question that way, then the answer will depend on how big the square and the circle are. If we’re talking about the specific picture on this page, then the difference can’t be bigger than the entire area of the page you’re looking at. But if we solved the same problem when the circle was as big as a planet, then the answer would be much bigger.

It would be nice if we could talk about the answer in a way that didn’t depend on how big the picture was. So what if we answer the question by giving the area of the circle divided by the area of the square? That’s no more or less correct than the other way, but at least we have the hope of just giving one answer. There’s got to be some number, right? The circle is clearly taking up more than 1% of the square, and it’s clearly taking up less than 99% of it, so there must be some number between 1% and 99% that represents the exact answer, and it should be the same answer whether the whole picture is tiny or huge, because the square and the circle shrink or expand together as we resize the picture.

Okay, well the circle is curvy, and the square isn’t, so we’re stuck. The infinite magnifying glass we invented helps us deal with curvy stuff, but the only thing we’ve used it for so far is figuring out the steepness of curvy things. This circle question doesn’t seem to be a question about steepness, so it’s not clear that derivatives would help. Let’s do something that won’t get us any less stuck, but it might help us to look at the question in a different way. Let’s break the square up into four pieces (see Figure 4.2).

Now we can rephrase the question as, “How many of the little squares would you need to make up the circle?” recognizing that we don’t necessarily expect the answer to be a whole number. Let’s call the area of one of the small squares , and the area of the big square . So we can write and. . . we’re stuck again. The lines didn’t make the problem any easier. We still don’t know the area of the circle because of those obnoxious curvy bits. But now we can talk about the problem in slightly different language. Just by staring at the picture we can tell that is bigger than one of the tiny squares, and it definitely looks bigger than two of them, so it’s almost certainly true that . It’s less clear whether or not the circle is bigger than three of them. If we had to guess at this point, we’d wager that the exact number was somewhere in the neighborhood of 3.

To be honest, we haven’t gotten anywhere, and we’re just goofing around. We’re still stuck, because we don’t know how to figure out the area of curvy things, including circles. So let’s cheat! What do I mean by cheat?

In the Molière play The Imaginary Invalid, a doctor is asked why opium puts people to sleep. He answers that the substance causes sleep because of its “virtus dormitiva,” a Latin term that means “sleep-inducing power.” Once we unwrap his terminology, we see that the doctor was clearly not answering the question. He just invented a fancy-sounding name for the ability of opium to put people to sleep, because he didn’t know the answer. That’s a ridiculous and irresponsible thing to do when you don’t know the answer to a question, so let’s do it!

Figure 4.2: That didn’t help much.

Our problem is to figure out how much of the square is filled up by the circle. We don’t know the answer, but there has to be some answer. That is, there has to be some number — let’s call it # — such that . Just from looking at the picture, we can confidently say that 2 < #< 4, but we don’t know exactly what this number # is. We also know that , so we can express the answer in a way that doesn’t depend on if we do this:

That may look fancy, but we still haven’t really done anything! We’re just Molièring the question by inventing a name for the thing we don’t know. In this case, our name was the symbol # instead of the fancy sounding “virtus dormitiva,” but there’s really no difference in the underlying method. We’re simply defining # and then giving up. Let’s summarize what we’ve done so far.

Question: How many of the little squares does the circle take up?

Answer: It takes up # of them.

Question: What number is #?

Answer: I don’t know. Leave me alone.

Notice that , where r is the distance from the center of the circle to its edge, or what textbooks call the “radius” of the circle (go make sure you see this). For the same reason, we know that . We defined # as whatever number makes true. But this says

This may remind you of the formu—

(Mathematics wanders into the chapter.)

Mathematics: WELL HELLO AGAIN, YOU TWO. WHAT’S ALL THIS?

Author: Not much. We’re stuck.

Mathematics: ON WHAT?

Reader: The problem above this.

Author: Go give it a read.

(Mathematics leaves, and returns after a short delay.)

Mathematics: OH. I SEE.

Reader: Any idea what we should do?

Mathematics: NO. WELL, NO HELPFUL IDEAS. I MAY HAVE A FEW UNHELPFUL ONES, THOUGH. THIS REMINDS ME OF A SIMILAR PROBLEM I WAS STUCK ON A FEW DAYS AGO.

Author: What kind of problem?

Mathematics: MIND IF I MAKE MY OWN SECTION?

Author: Go ahead.

Mathematics: THE PROBLEM IS AS FOLLOWS. . .