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Mathematics-Online course: Basic Mathematics - Real Numbers

# Rational Numbers

The rational numbers consist of signed fractions of integers:

where denotes the greatest common divisor. The set of rational numbers forms a field with the standard arithmetic operations (addition/subtravtion and multiplication/division).

As is illustrated in the figure, is countable. With the so-called diagonal method, we can enumerate all positive fractions. Uncancelled fractions are skipped in the count.

Finally, the rational numbers form a dense subset of the real line. Between any two different rational numbers there are infinitely many rational numbers.

(Authors: Höllig/Kimmerle/Abele)