Perri O'Shaughnessy - Case of Lies

To him these four numbers were as real as rocks, more real, alive in some sense. But what were they? What was a number? Where did numbers come from? Had humans invented them or discovered them? Where did they go? He thought they followed a line toward some far infinity where a little breeze sprang up and supported them.

He never saw the integers as hard-edged; to him they were like clouds, with moving centers depending on what was pulling on them from either side. The clouds touched each other, even early on in the line of numbers, but as the numbers got bigger the clouds became a continuum, a long streak of cirrus.

But when he was young, he had little interest in the large numbers; it was Zero, One, Two, Three, and the rest was just amplification.

“Yes, Elliott?” Mr. Pell said from the blackboard. Sharon, the girl in the desk next to him, grimaced, because Elliott was a pudgy pest who kept his hand raised all through class. He couldn’t help it. Mr. Pell kept saying all these things that made no sense. The class had spent most of the year memorizing the multiplication tables, which Elliott already knew, and this month they were learning long division.

“Why is the answer zero when you multiply by zero?” Elliott asked.

“Because that’s how the system works,” Mr. Pell said. Then he sighed and said, “Zero times three equals zero plus zero plus zero. Think about it.”

Elliott was supposed to be quiet now, but instead he argued, “But zero is nothing. It can’t do anything to another number. Three times nothing can’t change three.”

“No, one is the number that doesn’t change anything in multiplication,” Mr. Pell said.

“Then zero and one must be the same number,” Elliott told him. The class giggled, even Sharon, as if he had said something funny, but he felt a need to know, or maybe to be right, and it didn’t stop him. “Ten times one, that’s ten times itself, isn’t it? Shouldn’t that be a hundred? One should be a-a-”

“An exponent,” Mr. Pell said. “You’ll learn about them next year. Ten times itself is a hundred, true. But ten times one is ten.”

“And ten times nothing is nothing?”

“Good. Right.”

“Then what is ten divided by zero?”

Tall Mr. Pell looked at the big clock on the wall and finally said, “You can’t divide by zero, Elliott. It’s a rule.”

“Why is it a rule?”

“Because the rest of arithmetic won’t work otherwise. You just have to accept it.”

“I thought math was supposed to be logical.”

“It is.”

“Then how come multiplying by nothing wipes out a number?”

“Talk to me after class.”

Mr. Pell went back to the blackboard after the bell rang and the rest of the class ran out. He was awfully young to be a teacher. Elliott’s father said Mr. Pell had been a PE major, but he’d minored in math and the school needed a math teacher more than a coach. He wore a bow tie and a short-sleeved blue shirt. He looked like Eddie Murphy, but without the funny stuff.

“Look. Division is based on multiplication, right? Twelve divided by zero equals x. Then zero times x would have to equal twelve, but that can’t be. Zero times x equals zero, you already know that.”

“But why?”

“Because,” Mr. Pell said, “it works. A million math operations say it’s true. Let’s look at it this way. Let’s take nine divided by three. If you have nine rocks, you can separate them into three groups of three. Got that?”

“Sure.” In his mind they were reddish rocks, like on Mars.

“So let’s look at nine divided by zero. How many groups of zero can you separate nine rocks into?” Mr. Pell smiled. “You see? You can’t have a group made of nothing. It just doesn’t make sense.”

“That’s an artifact of your definition,” Elliott told him.

Mr. Pell dropped his chalk. “Who told you that?”

“It’s logic.”

The teacher gave Elliott a long look. He seemed excited. Elliott thought, I’m a bright one, and warm satisfaction spread through him. He couldn’t wait to see what Mr. Pell would come up with next. Without noticing, he had clenched his fists and stood with his legs apart, chin out.

“This isn’t a boxing match,” Mr. Pell said. “You’re pretty competitive, aren’t you? All right, Elliott. Let’s try looking at it this way. When you divide by a number, you expect the result to be a number. Got it?”

“Got it.”

“Let’s look at a sequence of numbers.” He wrote some fractions on the board. One over two, one over three, one over four, one over eight…

“See how the numbers change in a regular pattern? Get it?”

“Got it.”

“Know what happens if you keep on going this way?”

“They get smaller.”

“Very good! That’s right. The end result is something infinitely small. Approaching zero.”

“Awesome! It ends at zero?”

“No, it never ends.”

Elliott’s mouth fell open.

“It goes on forever, approaching closer and closer to zero. Zero is sort of the end of infinity.”

“So when it gets so small… when it’s one over zero… that’s infinity?”

“It’s something we simply can’t assign a number to at all. It’s outside the system. I’ll tell you why. You know what negative numbers are? Minus numbers?”

“Sure.”

“Try following another sequence: One over minus two, one over minus four, and so on. What’s at the end of the sequence?”

“Minus zero?”

“Good try. In fact, the answer is also zero. Because zero is zero. There cannot be a minus zero.”

“Why?”

“It’s not allowed. Don’t ask why. Just accept that the answer is zero for both sequences. But you can’t have the same answer for two different number sequences. Don’t ask why. You can’t. Since you can’t, we say that dividing by zero doesn’t result in a number.”

Mr. Pell expected Elliott to ask why you couldn’t have two separate answers, or why the second sequence was zero when it ought to be minus zero. He had a couple of slam-dunk sentences planned to put Elliott away, like “Don’t ask.”

But Elliott was way past that. “Yeah. That’s right. I always thought there was something strange about zero. Now I understand,” he said.

“Good.” Job well done, Mr. Pell’s face said.

“The number line must be a circle,” Elliott said. “Like a clock.”

“No. No.” The bell rang again and the next class started coming in and sitting down while Mr. Pell was still shaking his head.

“The number line. It’s really a circle. Like you said, the zero at both ends ties it together,” Elliott said hurriedly.

“No. The number line is a line. By definition.” But Mr. Pell rubbed his mouth and said, as if he were talking to himself, “… not bad. Sounds like elliptic geometry.”

“What?”

“Just accept that it’s a line, Elliott.”

“But why? Who made it that way? God?” Now several other kids were listening in. Elliott didn’t care. He needed a real answer, not an answer for a kid, an answer that worked for him, or else it might be that the nagging thought he sometimes had at night was true-that he wasn’t a bright one after all, he was just the pudgy pest of the class, too stupid to understand what was obvious to Mr. Pell.

If he couldn’t understand a simple thing like why you can’t divide by zero, then he’d never understand anything. He felt like he was going to bust out crying. Why couldn’t Mr. Pell answer the question in a way he could understand?

Elliott said loudly, “You don’t know anything, I guess,” to his teacher. He heard the laughing in the background again. Everybody thought he was a freak. It made him mad. “I know what an exponent is,” he boasted. “I know what a square root is. What’s the square root of minus one?”