Proof by contradictionTheorem: The reciprocal of any irrational number is irrational.
Lemma: If is rational, then is rational.
Proof: Let be rational. Then where are integers with and . Then . Thus, and since we see that . Thus, is rational.
Now let be irrational and assume by contradiction that is rational. Then by the Lemma, is rational. But, which means a rational number equals an irrational number, a contradiction. Thus, is irrational.