Evaluating magnitudes of non-algebraic numbers

**espen180** · Jun26-09, 04:51 PM

Just like it is possible to show that $LaTeX Code: e^\\pi > \\pi^e$ , is it possible to show that

$LaTeX Code: \\sqrt{2}^{\\sqrt{2}} > \\frac{1+\\sqrt{5}}{2}$

or

$LaTeX Code: \\sqrt{2}^{\\sqrt{2}} < \\sqrt{3}$

**mathman** · Jun26-09, 07:56 PM

Originally Posted by espen180

Just like it is possible to show that $LaTeX Code: e^\\pi > \\pi^e$ , is it possible to show that

$LaTeX Code: \\sqrt{2}^{\\sqrt{2}} > \\frac{1+\\sqrt{5}}{2}$

or

$LaTeX Code: \\sqrt{2}^{\\sqrt{2}} < \\sqrt{3}$

Since all these numbers are well defined, it is possible, although at times it might require a little work.