Doing math in school is usually about getting an answer. Doesn’t matter if you’re multiplying 3-digit numbers, integrating by parts, figuring out what happens when a train leaves Chicago at 6 AM, or counting paths through a maze. Doesn’t matter if you’re in grade school or high school, math class or math team. Problem. Answer. Find out if your answer was right. On to the next problem.
It’s a shame, because the first time I really thought hard about math was when this paradigm blew up. It happened around seventh grade, when we were learning how to convert repeating decimals into fractions. We had done a few standard problems (converting, say, 0.333… into 1/3, or maybe 0.424242… into 42/99, simplified to 14/33), when along came 0.999… Which, if you used the standard method (multiply by a power of 10, subtract the original from the result to cancel the infinitely repeating part, solve for the original), appeared to be equal to 1. And that was an answer I wasn’t remotely ready to accept.
Point (me): That couldn’t be right, because we started with 0-point-something, which is clearly less than 1.
Counterpoint (teacher): OK, but you had no problem turning 0.333… into 1/3, and we’re just repeating the same method when we turn 0.999… into 1.
Point: But I can turn 1/3 back into 0.333… by dividing 1 by 3, and I can’t turn 1 into 0.999… by dividing 1 by 1.
Counterpoint: You can. It’s just a funky kind of long division, with a remainder of 1 each time. At the first step, you’re dividing 1 by 1, and you say the answer is 0, remainder 1. From then on, you’re dividing 10 by 1, and you say the answer is 9, remainder 1, over and over again.
Point: That’s against the rules, the remainder has to be less than what you divide by!
Counterpoint: If you agree that 0.333… = 1/3, just multiply both sides by 3 and see what you get.
Point: But the decimal expansion of 1 is 1.000…! How can it have another one?
Back and forth we went (did I mention that I loved to argue?). It wasn’t question and answer anymore, but questions spawning questions. Was it really OK to say 10 times 0.999… was 9.999…, or were we pulling some strange extra little bit from infinity? It did look OK to multiply 0.333… by 10, but was that somehow suspect too? What was really going on out there at infinity, and what did it mean to be just a tiny smidgen less than 1? What were the real rules of long division, anyway?
For the first time, math seemed very open, up for grabs. My seventh grade mind wasn’t even sure what the answers to these questions could look like. My teachers said there were these things called limits, which helped you represent what happens out at infinity. So 0.999… wasn’t an ordinary number, it was a limit, but it was equal to 1, which was an ordinary number. My head spun. Eventually I declared that anything with infinitely repeating 9’s was undefined and called it a day. (That was right in a way: nobody defines infinite sums carefully in grade school, although I wouldn’t have said that, say, 0.333… was undefined too.) And everyone went back to the usual routine. Problem. Answer. On to the next problem.
But several aspects of the experience stayed with me to this day:
1. An expanded sense of what a math question could be, and what you could learn from it. Once in a while, you’ll hear kids complain about not understanding what a math problem means, or what it’s asking them to do. More frequently, at least these days, the parents are the ones complaining (and much more loudly, too). Often they’re right: hundreds of poorly written math problems get sent home every day. But sometimes it’s through trying to make sense of a question, whether someone else’s or your own, that you learn the most. Does 1 – 0.999… = 0? Why or why not? These were questions of a different kind, as far as you could get from the land of how many more marbles does Dorothy have than Fred. Behind the scenes, infinity was revealing itself, as the subject matter (where 0.999… finally reached 1, or didn’t), and also as the true scope of math. It was thrilling.
2. Years later, when I finally got to learn about limits, I paid a lot of attention. I’d been promised that they would resolve the mystery, and they did! Short summary in case you haven’t studied this stuff: the idea is that when you write down 0.999…, or any other decimal that doesn’t terminate, the dots mean that you’re not writing down an ordinary number in the literal sense. Instead, 0.999… is shorthand for a sequence of numbers (here 0.9, 0.99, 0.999, 0.9999, and so on), defined by some rule that pins down exactly what “and so on” means. (In this case, the rule is that you get the n-th element of the sequence by adding 9/(10n) to the n — 1-st element. For example, you start with 9/10 as the first element, add 9/100 to get the second, then add 9/1000 more to get the third.) Out at infinity, this sequence converges (gets arbitrarily close) to 1, meaning that you can’t squeeze any other number between 0.999… and 1. Once you have convergence, limit theory tells you that the arithmetic manipulations are OK: you’re allowed to write 3×0.333… = 0.999…, 10×0.999… = 9.999…, and so on. There’s a lot more to say here, and this wonderfully detailed yet accessible article by Jordan Ellenberg is very insightful on both the math and the underlying intuition.
It was amazing to me to learn all this. Back in seventh grade, I had gone from some ordinary-looking manipulations of sums and products to a set of questions that felt like philosophy, and now here was math providing real answers to those questions, on its own terms, putting me back on firm ground. In their way, the answers were as mind-expanding as the questions had been. How cool was that?
3. Speaking of those ordinary-looking manipulations, I would never naively trust them again. Maybe your eyes glazed over when you first encountered proofs in math class: why bother proving things that seem out-and-out obvious? But for me, after seventh grade, the obvious could be questionable (like those algebraic manipulations that led to such a weird outcome) or flat-out false (a single number really could have two separate decimal expansions). So when I finally got to proofs in school, I couldn’t have been happier. You mean you can actually prove it? You can carefully work through the all moving parts and identify the statements and methods that you can genuinely trust and use? Bring it on!
4. Faith in the long view. Once I heard someone say that math has two-minute problems, two-hour problems, two-day problems, two-month problems, and two-year problems. I wasn’t ready for multi-year territory as a seventh grader, already getting antsy after a couple weeks without a real answer. Being able to arrive at a satisfying solution eventually, years later, was a very big deal. For a long time afterwards, when I got stuck on a math problem, or on something else, I would recall how long it took to make sense of 0.999… And sometimes, with a little more work, a few hours or days or months later, I’d get unstuck. For what it’s worth, even this post took a few tries over a couple weeks to write.
So, kids, don’t be afraid of questions that might be a little unclear, or that don’t point you directly to an answer. And parents, don’t rush to ridicule that confusing math homework sheet on Facebook. Maybe a little confusion is part of the territory, and just means that you haven’t solved the problem yet. Not understanding can make you frustrated, but it can also mean you’re on the cusp of learning something. Or even that the learning is already underway.