There are infinitely many real numbers between 0 and 1. Are there twice as many between 0 and 2, or are the two amounts equal?

Question

I know the actual technical answer. I’m looking for a witty parallel that has a low chance of triggering an infinite “why?” procedure in a child.

In: 1335

Answer 1

To start off with, let’s talk about how mathematicians count things.

Think about what you do when you count. You probably do something like looking at one object and saying “One”, then the next and saying “Two”, and so on. Maybe you take some short cuts and count by fives, but fundamentally what you are doing is pairing up objects with whole numbers.

The thing is, you don’t even have to use whole numbers, pairing objects up with other objects also works as a way to count. In ancient times, before we had very many numbers, shepherds would count sheep using stones instead. They would keep a bag of stones next to the gate to the sheep enclosure, and in the morning as each sheep went through the gate to pasture, the shepherd would take a stone from the bag and put it in their pocket, pairing each sheep with a stone. Then, in the evening when the sheep were returning, as each one went back through the gate, the shepherd would return a stone to the bag. If all the sheep had gone through but the shepherd still had stones in his pocket, he knew there were sheep missing.

Mathematicians have a special name for this pairing up process, *bijection*, and using it is pretty important for answering questions like this, because it turns out using whole numbers doesn’t always work.

Now, let’s get back to your question, but we’re going to rephrase it. Can we create a bijection and pair up each number between 0 and 1 to a number between 0 and 2, without any left over?

We can, it turns out. One way is to just take a number between 0 and 1 and multiply it by two, giving you a number between 0 and 2 (or do things the other way around and divide by 2). If you’re a more visual person, [here’s another way to do this](https://i.imgur.com/exWVWkU.mp4). The top line has a length of one and the bottom line a length of two. The vertical line touches a point on each line, pairing them up, and notice that as it sweeps from one end to the other it touches every point on both lines, meaning there aren’t any unpaired numbers.