Talk:Gelfond–Schneider constant

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[edit] What's that nonconstructive proof thing really trying to say?

eln 2 after all is an irrational number to an irrational power. Phr 06:05, 18 February 2006 (UTC)

The square root mentioned in the article is irrational, but raised to the power √2 you get 2, which is rational. This shows that √2√2 is transcendental but seems to me like a constructive proof that an irrational to the power of an irrational can be rational, so I don't know what the nonconstructive proof referred to. Chenxlee (talk) 22:17, 6 February 2008 (UTC)
The Pythagoreans knew the square root of two was irrational. Suppose in 1901 you (or Hilbert) did not know whether \sqrt{2}^\sqrt{2} was rational or not: if it was rational then \left(\sqrt{2}\right)^{\left(\sqrt{2}\right)} would have been an example of an irrational to the power of an irrational being rational; if it was irrational then \left(\sqrt{2}^\sqrt{2}\right)^{\left(\sqrt{2}\right)} was an example of an irrational to the power of an irrational being rational. That was non-constructive as it depended on the status of a number without finding out that status. --Rumping (talk) 16:22, 9 June 2008 (UTC)