Prove that the set of all rational numbers is Countable.

To show that the set of all rational numbers is countable, we can use a simple trick: list them! We can express any rational number as a fraction \( \frac{p}{q} \), where \( p \) is an integer and \( q \) is a positive integer. By constructing a two-dimensional grid of integers, we can arrange the fractions systematically. Now, if we go diagonally through this grid, we can create a sequence that includes every possible rational number. We've got negative fractions, positive fractions, and even 0 (which is \( \frac{0}{1} \)). By skipping over duplicates (like \( \frac{1}{2} \) and \( \frac{2}{4} \)), we can ensure each rational number appears exactly once. Bingo! This shows that we can list them in such a way that they can be counted, proving that the set of rational numbers is indeed countable. Plus, numbering these bad boys means you can put your favorite rational number in a list—like a party invitation! Everyone's invited; just ensure no duplicates hop in the fun.