Last week we started working with infinite geometric series, a topic I personally love. First of all, it's one of the few places in a high school curriculum where deep, genuine philosophical questions bubble all the way up to the surface of a mathematical discussion. Second, it marks the place in my own academic life where I experienced a religious conversion to Orthodox Mathematicism:
In the beginning there was a single term. And to that term the Teacher did add another of smaller magnitude. Then a third term, smaller still, appeared upon the right hand side of the chalkboard, and it was revealed to me that the terms did decrease exponentially. My heart saw that this shrinking and adding proceedeth forever and ever, terms without end, Amen. And lo, when I beheld the sum, it was finite, and I knew that it was Good.
If my introduction to convergent series was a baptism, then using one to demonstrate that .999... = 1 was my confirmation. Now, having done the same thing with my students, I think it might be even more interesting from this side of the desk. In particular, two of their questions/comments highlight two very different understandings of infinity and the real numbers.
First, the ingredients of a metaphor. If you've ever been a runner, this is easy. If not, I'm going to need you to go on a quick jog before you read any farther so you can appreciate the rest of this carefully crafted rhetorical device. I'll wait...
When you drive the same stretch of road over and over again, you tend to experience it dynamically. You pass a landmark, anticipate a curve, accelerate over a little rise. The road changes in front of your eyes. You see the road as a process. But when you run along the same route, it looks completely different. There is just this monolithic expanse of concrete laid out over the landscape. You can creep around and explore its different features, but you experience the road essentially as a static object. In other words, you experience the road as it actually is. Keep this in mind as you read the following two questions from my actual students.
D: "But Mr. Lusto, if .999... is exactly 1, then .999... plus .999... should equal exactly 2, but it doesn't. It's 1.999...8."
What a freaking fantastic argument! Here's a student who has accepted my proof, interpreted it, thought about it critically, and deduced a logical contradiction. My heart swelled a little bit. Unfortunately, the flaw in his reasoning highlights a fundamental misconception. D is viewing .999... like a driver. He sees it as a dynamic process, repeatedly appending a 9 to an ever-expanding sequence of 9s. He might even accept that this can theoretically go on forever, but his point-of-view still gets him into some trouble. When D mentally sums .999... and .999..., he's suggesting that there are two "last 9s" that, when added, produce a trailing 8. But of course there are no "last 9s." He's implicitly terminated the process prematurely (which is to say, at all). Hence his objection, though thoroughly beautiful, is ultimately illusory.
J: "But Mr. Lusto, if .999... equals 1, then doesn't 1.999... equal 2? Then can't we write every number in two different ways?"
This student views .999... like a runner. The reason that .999... and 1 can be meaningfully thought of as equal is because they represent the same static value. They're just two different names for the same object. Here's a student who sees .999... as it actually is. And now, because of that, his concern is genuine. The fact that many real numbers have two decimal representations (one with infinite trailing 0s, one with infinite trailing 9s) is a true mathematical/philosophical problem. In fact, it's an important result: those sorts of numbers turn out to be dense in the reals (in the topological sense). J may never care about, or even get enough math under his belt to understand, that statement, but his view of the nature of infinity is already more nuanced than D's.
Something to think about next time you're driving. Better yet, next time you're running.