7 - Numeration Systems

Summary

“You boys have been swallowing a lot of mathematics lately, haven't you?” Alice said.

“Indeed we have,” said Tweedledum.

“And we have digested most of it,” said Tweedledee. “We only have to take a few indigestion pills now and then. On the whole, I think we have come a long way.”

“Good,” said Alice. “It does take time to ruminate, as the holy cow may tell you. Nevertheless, perhaps it is time for me to give you an examination.”

“Don't!” said the twins together. “We don't like examinations.”

“I understand how you feel,” said Alice sympathetically. “The Red Queen gave me an oral examination in arithmetic once, and I failed.”

“Do you remember any of the questions?” asked Tweedledum.

“There was only one question,” said Alice. “It was kind of long. So here it goes. A stagecoach went from London to Harwich and started out with six passengers. Do you think you can remember that?”

“Of course I can remember that,” replied Tweedledee. “There's not much to remember!”

“Very well,” replied Alice, “the coach made a stop. Two passengers got off and four passengers got on. Got that?”

“Yes,” replied Tweedledum, who was keeping count.

“Then the coach went on and made another stop. Three passengers got off. Are you following?”

“Yes,” said Tweedledee, who was faithfully keeping count.

“Then the coach went on and made another stop. Two passengers got off and two passengers got on.”

“That's the same as if the coach hadn't stopped at all!” exclaimed Tweedledee.

“Anyway,” continued Alice, “the coach went on and made another stop. Three passengers got off and five passengers got on. Are you still keeping count?”

“Yes, I am,” said Tweedledum.

“Then the coach arrived in Harwich. All the passengers got off. How many times did the coach stop?”

“Oh,” swallowed the twins. “We weren't counting that!”

“You'll never be able to pass an examination unless you can count,” said Alice.

“But we can count,” pleaded the twins together. “It's just that we counted the wrong thing!”

“That's no excuse,” replied Alice. “You should always count everything, because everything counts.”