Why are the rational numbers countable while the real numbers are not countable?

In mathematics, the distinction between countable and uncountable sets lies in whether their elements can be placed into a one-to-one correspondence with the natural numbers.12

Countability of Rational Numbers

A set is considered countably infinite if its elements can be listed in a sequence, even if that list is endless. The rational numbers—numbers that can be expressed as fractions—are countable because they can be systematically arranged and paired off with the integers without any elements left over.312

Uncountability of Real Numbers

In contrast, the real numbers are uncountable. Georg Cantor proved this using his famous diagonalization argument, which demonstrates that it is impossible to create a complete list of all real numbers that corresponds to the natural numbers. Because the real numbers cannot be paired with the integers in this way, they represent a "larger" infinity, often referred to as the continuum (c).142

While both sets are infinite, the infinity of the real numbers is strictly greater than the infinity of the rational numbers.52