“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?”

“This is way beyond third-grade arithmetic,” Mr. Pell said. “Who told you to ask me these questions?” He still had a peculiar look, like he was really interested, too, and this emboldened Elliott.

“Nobody. My pop. He’s a Sanskrit scholar. What’s the square root of minus one?”

“You know what? I bet your father already told you the answer, told you it’s an imaginary number with its own number line.”

“Egg-zackly. So if you can set up a brand-new number line for negative square roots, why can’t you set up a new number line for one divided by zero?”

Mr. Pell looked down at him from the height of a mountain. He bent and picked up the chalk, and said in a low tone, close to Elliott’s ear, “Listen to me, kid. I’m going to tell you a secret. You’ll understand it better when you’re older. Do not tell the rest of the kids about this. You cause enough trouble already. They’ll get confused.”

Elliott raised his eyebrows. He tried to look nonchalant.

“You can divide by zero, if you invent another arithmetic. This arithmetic you’re learning-it’s just the one that works best for things like building houses. There are all kinds of arithmetics and geometries.”

Elliott understood immediately. His head swam. The relief was so overwhelming, he almost fell down. This arithmetic was a game, and there were other games.

“Got it?” Mr. Pell said. “Satisfied? Now beat it, would you? Please?”

On his tenth birthday, his father gave Elliott an old edition of Euclid ’s Elements. Winter had brought its cold wind to sweep down on the island. Elliott stayed up in his room for two weeks. When he came down he said, “I don’t understand this at all.”

“Let’s have a look.” They opened the book to Euclid ’s assumptions, the logical statements that are self-evident and are the basis of plane geometry.

“Two points make a line,” the book said.

“Why?” Elliott said. “The line could stop halfway to the second point. Or the two points could be on top of each other, so it looks like one point. Or the line could be wavy.”

“Oh, I quite agree. But you have to think like Euclid,” his father said. He smoked Marlboros. The smell of math to Elliott forever more would be connected to the smell of burning tobacco. They were in Pop’s warm den, piles of papers and books everywhere, the TV on a football game as usual. Elliott’s mother was sitting on the chair under the window, reading a book, her brown hair lit by the lamp.

“ Euclid developed a system that hangs together, that’s the main thing. Let’s try to make his sentence about points more accurate. He’s saying that if you take any two points in the universe, the simplest relations between them is generally a directional arrow that we call a line.”

“Okay. That makes sense. But why triangles? Angles and sides and all that. Why is a right triangle so important?”

Pop stubbed out one cigarette and fired up another one. “Because the Greeks discovered that they could say beautiful, simple, elegant things about right triangles. And because they could build houses using right triangles.”

“Houses again! How come it’s always about houses? Why not start with a-a cloud? Why not invent a formula for finding the volume of a cloud?”

“Too messy,” his father said. “ Euclid started with something easy and useful. In all fairness, he was fond of squares and circles too.”

“Why did he get to make up his own rules? They’re wrong!”

“The one about parallel lines may not always work. The others have stood up pretty well,” Pop said mildly.

“But what about two points making a line? I could make a system where they don’t, couldn’t I?”

“Attack the system at your own risk. I’m going to tell you a story.” Commercials had taken the place of football on the TV in the wall unit across from his father’s desk. He muted the sound and said, “A long time ago there was a genius named Pythagoras. He was a genius because he made some discoveries about the integers that no one had ever made before. These discoveries were so elegant, so incredible, that numbers became a religion. The Pythagoreans believed, for instance, that the cosmos formed from a one. It split into the integers, which formed themselves into geometrical shapes, and finally became air, earth, fire, and water. All Nature, all Reality, grew from Number.”