# Matching Infinite Sets

By mariam zohdi

Topic: Maths

[Open the interactive Jig](https://ask.jig.com/j/urNIPt3bY4nd)

## Guide

### 1. Pair them up

How can you tell two groups are the same size without counting? Pair them up\. If every marble gets exactly one partner and nobody is left over, the groups are the same size\.

Mathematicians call a perfect pairing like this a one\-to\-one matching\.

### 2. One left over

Now a sixth blue marble rolls in\. It has no partner, so the blue row is bigger\. For groups that end, pairing and counting always agree\.

### 3. Rows that never end

Now imagine rows that go on forever\. The blue row holds every counting number: 1, 2, 3 and on and on\. The gold row holds only the even numbers, each one sitting in front of its twin\.

Neither row has a last marble\. The rows simply fade into the distance\.

### 4. Surely half as many?

Look at the gaps\. In front of every odd number, like 1, 3 and 5, there is an empty space\. From 1 to 10 there are ten counting numbers but only five even ones, so the evens seem to be half as many\.

### 5. Slide the evens together

Press the button to slide the gold marbles together and close every gap\. Each even number lands in front of its half: 2 with 1, 4 with 2, 6 with 3\. The rule is simple: every counting number is paired with its double\.

### 6. Nobody left over

Every counting number has exactly one partner: its double\. Every even number has exactly one partner too: its half\. Nobody is left over in either row, so these two endless rows are the same size\.

Tap any marble to light up its partner\. Even 1,000,000 has one: 2,000,000\.

### 7. The endless hotel

Picture a hotel with endless rooms, every one full\. Each guest moves to the room with double their number: room 1 to room 2, room 3 to room 6\. Now every odd room is empty, ready for endlessly many new guests\.

The mathematician David Hilbert described this hotel in a lecture in 1924\.

### 8. Galileo’s puzzle, Cantor’s answer

In 1638 Galileo paired numbers with their squares and was puzzled\. Later, Georg Cantor made matching the rule: if a perfect pairing exists, two sets are the same size, even endless ones\. He also proved that some infinities, like all the points on a line, are too big to pair with 1, 2, 3\.