4.2.2. Set of Rational Numbers

Now, we will see next whether the set of rational numbers which I denote by this ℚ notation is countable or not. Now intuitively it might look the answer is no because definitely rational numbers is a super set of the set of the integers. And looks like there is no way of sequencing because the fundamental fact about rational numbers is that you take any 2 rational numbers there are infinitely many more rational numbers between the same 2 rational numbers.

Rational numbers (Q) are numbers that can be expressed as a fraction p/q, where p and q are integers, and q is not zero. Initially, one might think that because there are infinitely many rational numbers between any two rational numbers, it's impossible to find a way to list them all. This creates an impression that the set of rational numbers might not be countable. However, that's a misconception, as we will see.

Imagine a cake that is infinitely layered. If you take any two slices, you can always find another slice between them. This makes it seem like you can't count the slices. But if we organized them wisely, we could still list all the slices one by one!

But what we can do is we can show a very clever enumeration of the set of rational numbers which will prove that the set of rational numbers is a countable set. The sequencing that we are going to see here will be based on the sequencing of the elements of the point in the 2 dimensional integer plane.

To show that the rational numbers are countable, we can use a systematic enumeration method based on integer coordinates in a two-dimensional plane. If we consider each rational number as a point (p, q) in this plane, where p and q are integers and q is not zero, we can travel through this plane in a structured way (like a spiral or grid pattern) to list out all possible rational numbers. This method ensures that we don't miss any rational number in the process.

Think of a giant chessboard where each square represents a rational number. If you start at one square and move in a spiral, you systematically touch every square, ensuring that all rational numbers are counted without any being skipped over.

The idea is if we consider any rational number and if it is a rational number it will be of the form p / q, where p is some integer and q is some integer and q will not be 0. So the idea is traverse or you follow the enumeration of all the elements in the set ℤ x ℤ namely this enumeration here.

By taking points in the integer coordinate system, we check each (p, q) to see if q is not zero. For each valid pair, we correspond it to a rational number p/q. We systematically move through the grid using a structured approach so that we can check every possible rational number, ensuring each is listed without missing any.

Imagine navigating a crowded city with many crossroads, like a grid. If you make sure to visit every intersection systematically, you will eventually come across every street, just like we ensure every rational number is counted.

So now you can see even though there are infinitely many rational numbers if I follow these 2 rules of enumerating the rational numbers I will not be missing any rational number because you take any rational number it will be of the form (p / q). And if will be eventually listed down in the sequencing that I have specified here.

By following the established rules to move through the coordinate grid, we ensure every rational number of the form p/q is eventually included in our enumeration. Even though the rational numbers are infinite, our systematic approach guarantees we identify each one without repetition or omission.

Think of a librarian who knows to check every shelf in a huge library. Though there are countless books (rational numbers), by following a precise path through the aisles, the librarian can be sure to see every single book eventually.

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