Theorem: There exist irrationals x, y such that x^y is rational. Proof: Suppose that we already know that sqrt(2) is irrational. If sqrt(2)^sqrt(2) is rational, then the proof is done. If not (i.e. sqrt(2)^sqrt(2) is irrational), then (sqrt(2)^sqrt(2))^sqrt(2) = 2, a rational. Hope this helps. Warut On Mon, Nov 14, 2011 at 3:04 AM, Marc LeBrun <mlb@well.com> wrote:
Could anyone supply me with elementary examples that illustrate the idea of a non-constructive proof, for those with a "Martin Gardner reader" level of mathematical sophistication that also has a not-too-trivial but reasonably easily-verified case? For example a non-constructive proof that some set of numbers is non-empty, along with an example of such a number that can be fairly easily checked? Thanks!
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