There are fewer rational numbers than irrational numbers?

Answer 1

We can categorize all infinite sets into two categories. The first set is made up of a an infinite number of countable elements and the second consists of a infinite number of elements that cannot be counted. The set of rational numbers is countably infinite. This comes from that fact that we can easily count integers and natural numbers. Just remember to think of rationals as a ratio of two integers, a/b. The set of Irrational Numbers is uncountably infinite. There is no way to find a correspondence between irrational numbers and integers. By the definition of irrational, they can't be written as fractions. The fact that the natural numbers and rational numbers can be counted and the irrationals cannot be gives some intuitive understanding of why the latter is bigger.

In general, it turns out that an countably infinite set is smaller.

The German mathematician Georg Cantor did extensive work on the size ( cardinality) of sets including the these two. He provided some amazing proofs in two papers 1895 and 1897 and titled 'Beiträge zur Begründung der transfiniten Mengenlehre'

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This is an interesting topic and a link is attached to help the reader understand it better. The proofs require a good deal of set theory background So I have not presented them here, but will give link to those too.