The cardinal number of the set of rational numbers is aleph-null (ℵ₀).
This means that the set of rational numbers has the same cardinality as the set of natural numbers. Although the rational numbers seem much more numerous than the natural numbers, mathematicians have proven that it's possible to create a one-to-one correspondence between them. This is done through a process called countable infinity.
This means that we can list all the rational numbers in a sequence, even though there are infinitely many of them.
Here's a simple way to visualize this:
- Imagine a grid where the rows represent the numerators of rational numbers and the columns represent the denominators.
- We can then list all the rational numbers by moving diagonally through the grid, skipping any repeats.
While the set of rational numbers is infinite, it is still considered countable because we can put them in a one-to-one correspondence with the natural numbers.
