Countable set
The rational numbers are also countable, but its counting becomes tricky. We can't just list the fractions as (0/1, 1/1, -1/1, 2/1, -2/1...) since we will never get to 1/2 this way. However, we can list all of the rational numbers in the form of a table. The horizontal rows are the numerators, while the vertical columns are the denominators. We can then zigzag through the list, starting at 1/1, then 2/1, then 1/2, then 1/3, then 2/2, then 3/1, then 4/1, then 3/2, then 2/3, etc.  If we keep this up, we will get to all of the numbers on the table in due time. Each time we hit a new rational number that is new, we add it to the list of counted numbers. We don't want to count numbers twice, so when we hit 3/6, for example, we can skip it, because we already counted the rational number 1/2. In this way, we produce an infinite list with all the rational numbers. Therefore, the rational numbers are countable.