The irrational numbers are dense

Task number: 2830

Show that the set of irrational numbers is dense in the set of the real numbers, i.e. that in every non-empty open interval it is possible to find an irrational number. You may use the fact that the set of rational numbers is dense in the reals.

  • Resolution

    It suffices to solve the exercise in a bounded interval \((a,b)\), because every unbounded interval contains a bounded subinterval. Using the density of the rational numbers, we know that there is some rational number \(q\) in the interval \((a - \sqrt 2, b - \sqrt 2)\). Then \(q + \sqrt 2\) is an irrational number in the interval \((a, b)\).

Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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