As we all know, the points representing positive integers on the number line are distributed at regular intervals, and any two rational numbers can always be separated by another rational number, such as their average. This seems to suggest that there are far more rational numbers than positive integers. However, there are actually just as many rational numbers as positive integers. In other words, all rational numbers can be put into a one-to-one correspondence with positive integers.
This fact was discovered by Cantor in 1873. He first considered all positive rational numbers. According to the arrangement given below, we can establish a one-to-one correspondence between all positive rational numbers and positive integers. Obviously, every positive rational number appears in this array. If we rearrange them in turn according to the arrows and delete the numbers that have already appeared, we will obtain an infinite sequence of all positive rational numbers. Thus, there is an infinite sequence of all rational numbers. Therefore, a one-to-one correspondence between the set of rational numbers and the set of positive integers has been established, so there are just as many rational numbers as positive integers.