# 7 Days of Christmas: Day 6

For day 6 I will talk about how there are as many even numbers as natural numbers.

Firstly, the natural numbers are all positive integers. There are an infinite number of natural numbers. Also, there are an infinite number of even numbers. You might imagine that there are more natural numbers than even numbers because the natural numbers are made up of both even and odd numbers. However, this is wrong.

If we set up a one-to-one correspondence between the natural numbers and the even numbers, it shows that every even number can be matched to a natural number; every natural number has a number twice as large as it, and every even number has a natural number that is half its size.

This means that both infinite sets are the same size, which is called ‘countably infinite’. This distinguishes it from sets that are ‘uncountably infinite’, like the real numbers or the complex numbers as you can’t set up a one-to-one correspondence between the natural numbers and these sets.