6 - Rational and Irrational Numbers

Summary

Tweedledum and Tweedledee invited Alice for a treat. They put eight tarts on the table. They looked so deliciously inviting that Alice picked one up and started munching right away.

“Where did you boys get these tarts?” asked Alice, suddenly suspicious.

“You don't need to know,” said Tweedledum.

“We bought them from the Duchess's Cook,” said Tweedledee.

“You haven't got the money. You boys stole them from the Queen of Hearts.”

“She has got plenty, and won't miss a few,” said Tweedledum.

“Put them back,” ordered Alice, “or you boys will be in big trouble.”

“We can't,” said Tweedledee. “It wasn't easy getting into the Queen's pantry, and if we go back, we are sure to be caught this time. However, nobody knows anything so far, except you. You are not going to tell on us, are you?”

“Oh dear,” said Alice. “I don't suppose I can now, seeing that I have finished eating one of them already. So I won't tell, but promise me that there won't be a next time.”

“We promise,” the twins said together. “Let us share the tarts among us.”

“Fairly, I suppose,” said Alice.

“Of course,” said Tweedledum.

“If I remember correctly,” said Tweedledee, “we each get two, with two left over.”

“Not this time,” said Alice. “We better not leave any remainder for the Queen to find.”

“What is a fair share then?” asked the twins.

Fractions

If we share eight tarts fairly among three people and leave no remainders, each fair share is more than two tarts but less than three. Let us focus on just one tart at a time. To share it fairly among three people, it must be cut up into three equal pieces, and each person would get one of them. The amount of each share is to be represented by a new number such that the sum of three copies of it is equal to 1. We call this number one-third and write it as ⅓. With eight tarts, each will get 8/3 tarts.

In general, if the divisor is a positive integer n, each fair share amounts to 1/n. This number is called the reciprocal of n. We could also have called it the multiplicative inverse of n, because the product of n and 1/n is equal to the multiplicative identity 1.