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How can all fractions be arranged in a list with the natural numbers?

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Infinity — infinity Mathematical symbol representing the concept of infinity.

The set of all rational numbers—which includes all fractions—can be arranged in a list with the natural numbers because they are countable, or denumerable.12

This means there exists a one-to-one correspondence between the set of integers and the set of rational numbers, allowing every fraction to be paired with a unique integer. While it may seem counterintuitive that the set of fractions is the same size as the set of counting numbers, the German mathematician Georg Cantor rigorously demonstrated this property in the late 19th century.21

This countability stands in stark contrast to the set of real numbers, which Cantor proved is uncountable and represents a strictly larger infinity than that of the rational numbers.312