Friday, November 29, 2019

Real Numbers to Irrational Powers



In a related question we discussed raising numbers to powers.



I am interested if anybody knows any results for raising numbers to irrational powers.



For instance, we can easily show that there exists an irrational number raised to an irrational power such that the result is a rational number. Observe 22. Since we do not know if 22 is rational or not, there are two cases.




  1. 22 is rational, and we are finished.



  2. 22 is irrational, but if we raise it by 2 again, we can see that
    (22)2=222=22=2.




Either way, we have shown that there exists an irrational number raised to an irrational power such that the result is rational.



Can more be said about raising irrational numbers to irrational powers?


Answer



Some examples:





  • Eulers Identity eiπ+1=0


  • Gelfond-Schneider Constant: 22=2.66514414269022518865 is trancendental by Gelfond-Schneider.


  • ii=0.2078795763507619085469556198 is also trancendental


  • Ramanujan constant: eπ163=62537412640768743.999999999999250072597.


  • eγ where γ is the Euler–Mascheroni constant. (okay nobody has proved it's irrational yet, but surely is)



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