In Calculus class, it was often hard to resist distractions. I’d taught those kids for four years, and there was too much fondness built up between us. We swapped movie suggestions, debated Beatles albums. I’m surprised we got anything done.
But one day’s digression stayed surprisingly on-topic. (Well, at least it had to do with math.) The students got curious about repeating decimals.
Adrian led a pack of disbelievers in the claim that 0.999… = 1. (By 0.999…, I mean the decimal number in which the 9’s go on forever and ever.) “You’re crazy,” Adrian said. “You’ve lost your grip on reality, Orlin. Snap out of it.”
With my sanity under challenge (for neither the first nor the last time), I pushed back by offering the standard proof of the fact:
At this point, dissent began to flow among Adrian’s ranks. Some surrendered to my argument, but Adrian held firm. “Witchcraft,” he said.
“Fine then,” I said. “Give me a proof that they’re different.”
“No proof needed,” Adrian reported. “It’s by definition.”
“Ah,” I said, “but you wouldn’t want your definition to lead to inconsistencies, would you?”
“It doesn’t. Besides,” Adrian huffed, folding his arms, “you don’t tell me what I want.”
“Well, consider this,” I said, and quickly offered an alternative justification – not quite a proof – of the same fact, one to highlight the absurdity of Adrian’s stance:
By now, the other students had begun to turn on Adrian, who still held firm. “He must have clicked a YouTube video about 9/11 being an inside job,” Angel suggested. “He’s grown paranoid and delusional.”
“You’re delusional,” Adrian replied. “I’m just standing up for what’s right. And true. It’s the American way.”
I could have left it there – Adrian had delivered his performance, landed some good-one liners (even Kevin was cracking smiles), and instigated a debate that led to a cool proof. It was probably time to steer the lesson out of the wilderness and back onto dry pavement.
But the Socratic in me (I grew up with a family cat named Socrates) couldn’t resist another attempt. “Suppose you’re right, Adrian,” I said.
“Thank you.” Adrian bowed to the class.
I pressed on. “If 0.999… isn’t equal to 1, then what is it?”
“It’s the number right next to 1.”
“What do you mean?”
“It’s so close to 1 that there’s no number in between. It’s the biggest number that’s still smaller than 1.”
“I see,” I said. “So what’s the distance between 1 and 0.999…?”
“It’s a tiny little number, of course,” Adrian replied.
“The tiniest number of them all.”
“Isn’t that just 0?” I asked.
“There you go again, with your crazy talk,” Adrian scolded. “It’s not 0. That’s the smallest number. This is the second-smallest number.”
“And this second-smallest number,” I said, “what happens if you divide it in half?”
Adrian shook his head. “That’s dangerous. Like splitting the atom. Creates an explosion.”
“C’mon,” I said. “What if you take your little number, and divide it by two? Dividing it by two should make it smaller. But you said it’s already the smallest.”
“That’s why you shouldn’t divide it,” Adrian said. “You’re being reckless.”
“But can’t any number be divided by two? Isn’t there always a smaller number?”
He shook his head. “Nope.”
I turned to the rest of the class. “Does this remind you guys of anything?”
They nodded. “Limits.”
It was a teacher’s sweetest dream: an oddball digression that winds up shedding new light on the central themes of the course. I gestured triumphantly. “You see, Adrian? If there were a ‘smallest’ number, the concept of a limit wouldn’t make any sense. Calculus would fall apart.”
“No calculus?” Annie exclaimed. “You’re saying if we agree with Adrian, then calculus goes away?”
“All right!” Stephany said.
“Okay,” I said, sensing trouble. “I think it’s time we get back to Newton’s Method.”
“Why should we bother?” Jose said. “You just admitted that calculus doesn’t exist.”
Adrian was seizing up with giggles. “Is this funny for you, Adrian?” I asked.
He was too busy laughing to reply.
Note: As usual, I’ve used imagination to plug the holes in memory – though Adrian is every bit as lovable and mischievous as described. Good luck at SLO!
For what it’s worth, the notion of a “smallest positive number” (call it ε) wreaks havoc on the real number system. ε winds up behaving like a bizarre zero (bizero?). Specifically, the only way to salvage the existence of division is to say Aε = ε for any A (just like zero), and yet A + ε ≠ A (unlike zero). It creates a version of the reals where every number has precisely two “neighbors” – one directly above, and one directly below. And the distance to these neighbors is incomparably smaller than the distance to any other number.
Mathematicians out there: While this obviously destroys many of the reals’ nicest properties (ε, for example, has no multiplicative inverse, so we’ve lost field structure), are there any internal inconsistencies?
EDIT 9/7/2013: Check out James Tanton’s great take on this question.