How to prove that there are as many rational numbers as natural numbers? Why are irrational numbers more than rational numbers?

How to prove that there are as many rational numbers as natural numbers? Why are irrational numbers more than rational numbers?

The first problem is to establish a one-to-one correspondence between the natural number and the rational number is very easy to establish. For example, after arranging rational numbers according to Farey series, such correspondence can be established. The second problem can be explained by Lebesgue measure, because all rational numbers in [0,1] are countable, so the measure of the set composed of these rational numbers is 0, so the measure of the set composed of all irrational numbers in [0,1] is 1, which shows that irrational numbers are far redundant.

It is proved that the sum of a rational number and an irrational number is an irrational number

Let a=p/q (p, q are integers with prime) be rational and b be irrational.
Suppose that c=a+b is a rational number, let c=r/s (r, s are integers and prime)
So b=c-a=r/s-p/q=(qr-ps)/(sq) is a rational number.

Let a=p/q (p, q are integers with prime) be rational and b be irrational.
Suppose that c=a+b is a rational number, let c=r/s (r, s are integers and prime)
So b=c-a=r/s-p/q=(qr-ps)/(sq) is a rational number. Contradiction!