In matching the rational numbers with the natural numbers as we did in Example. what ratio...

In matching the rational numbers with the natural numbers as we did in Example. what rational number matches with 7?

Show that the set of rational numbers is countable.

SOLUTION:

First consider the positive rational numbers as we list them in Figure 2.18. The first row of the arrangement has all positive rational numbers with denominator 1. The second row has all denominators 2, the third row has denominators of 3, and so on. We then trace through the arrangement following the red line in Figure 2.18, skipping over numbers that we have encountered earlier.

We will follow the path through the rational numbers in Figure and list those numbers in a straight line, matching them with the natural numbers as follows:

Notice that in our matching, we have skipped over rational numbers that we have encountered earlier, such as 2/2, 2/4, 3/3, and 4/2.

When we look at the positive rational numbers in this way, we see that there is a first number, then a second, then a third, and so on, and we are listing each number exactly once. Therefore, there is a one-to-one correspondence between N and the positive rational numbers.

To show that the entire set of rational numbers is countable, we would argue as we did in Example to account for zero and the negative rationals; however, we will not go into the details of how to do it. Therefore, the set of rational numbers has the same cardinal number as the natural numbers, which is ℵ0.