Mirror and Tug

"That's a number line," Mirror said, smiling.

"That's a number line that drains and refills," Tug said. "Twice a day. Forever."

A small group of children were playing on the dock behind them. They were daring each other to walk down the wooden ladder that ran from the dock to the water. The ladder had rungs at one-foot intervals. The top rung was plus-five relative to the standard low-water mark. The bottom rung was minus-three — three feet below the lowest standard tide line, which meant it was always underwater. Even at the deepest tide, the bottom three rungs were submerged.

One of the children, a girl of about ten, was halfway down the ladder. The water was at plus-three. She was perched on the plus-four rung. Two of the lower rungs — plus-one and plus-two — were dry. Plus-three was just at water level. Plus-zero and below were underwater.

"How many dry rungs are there below me?" she called up.

Her brother counted. "Two dry," he said. "Three under water. Then it goes negative."

"How many total rungs to the bottom?"

"Top of ladder is plus-five. Bottom of ladder is negative-three. So... eight rungs total. Five above zero, three below zero."

"Eight!" she shouted up. "Including the wet ones."

Mirror leaned over to Tug. "Pretty good," she said.

"Wait for the next part," Tug said.

The brother on the dock yelled down: "But if the tide goes to negative-two, how many dry rungs will there be?"

The girl on the ladder thought. Then she said, very confidently: "Five plus two is seven. Seven dry rungs. The water will drop two more feet."

"That's right!" her brother shouted.

Mirror grinned at Tug. "She used negative numbers without flinching."

"She used negative numbers because the harbour uses negative numbers," Tug said. "Every kid in this town learns to read the tide-stains by age six. They know the water can be at plus-three, or zero, or negative-two. They know what those mean physically. The math comes later. The physical picture comes first."

"My father used to say that, too," Mirror said. "He'd hold up a glass sheet and a finished silvered mirror. He'd say: 'This is plus-one. That is negative-one. Same distance from zero. Opposite sides. One you look through. One you look at.'"

Tug nodded.

"That's the cleanest definition I've ever heard," he said.

A new problem broke out behind them.

A different child — a small boy of about eight — was standing on the plus-two rung. He was holding a small wooden boat. He wanted to lower it into the water without dropping it. The water was at plus-three. He needed to lower the boat down to the water, which was one foot below his rung.

He shouted up: "Dad! How far down do I lower it?"

The father, who was on the dock, called down: "The water is at plus-three. You are at plus-two. The water is above your feet."

The boy looked confused.

"Hold on," Mirror said, standing up. "Let me try this one."

She walked over to the edge of the dock and leaned down so the boy could see her. "Hello," she said.

"Hello," the boy said.

"You're at the plus-two rung. The water is at plus-three. Where is the water compared to your rung?"

The boy thought. "Above me?"

"Above you, by how much?"

"One foot."

"Right. So to put the boat into the water, you have to lift the boat up by one foot. Not down. The water's above you."

The boy looked at the water, which was indeed sloshing just one foot above his rung. He laughed. He climbed up one rung. He set the boat gently onto the water. The boat floated away.

The father shouted thanks.

Tug, watching, said quietly: "That was the inverse move. He thought he needed to lower. You showed him he needed to lift. Same distance, opposite direction. That's all an inverse operation is."

Mirror sat back down beside him. "It's the same as the mirror thing," she said. "The reflection. Lower-by-one is the opposite-direction-by-one of lift-by-one. The boy needed to do the reflection. He was about to do the wrong half of the pair."

"That's the whole inverse-operation principle in one boat," Tug said. "If you go the wrong direction by one foot, you correct it by going the opposite direction by one foot. Minus and plus are mirrors of each other. They are the same magnitude, opposite sign. They undo each other. Tug and counter-tug."

He paused.

"You know," he said, "I've been teaching inverse operations for fifteen years and I have never thought about them as mirror images before. I always thought of them as pulls in opposite directions. But they're the same idea. The mirror image of plus-five across zero is negative-five. The inverse of adding five is subtracting five. Same symmetry. Different vocabulary."