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November 28th, 2012, 12:02 PM  #1 
Member Joined: Oct 2012 Posts: 36 Thanks: 0  complex number calculation
Evaluate {(1+i)^(1i)} and describe the set{1^x} when x is a real number, distinguish between the cases when x is rational and when x is rational. for now considering the complex number. i dont know how to start with,for firest part i just write it into e^((1i)log(1+i)) then get the number with e to the power which inculding i , and the secound part , for 1=e^(i2npi) then 1^x is e^(2inxpi) then how to consider the case for rational and irrational here?????

November 28th, 2012, 03:28 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,641 Thanks: 625  Re: complex number calculation
When x is rational there are only a finite number of solutions, while for x irrational there are an infinite number of solutions.

November 28th, 2012, 03:44 PM  #3  
Member Joined: Oct 2012 Posts: 36 Thanks: 0  Re: complex number calculation Quote:
 
November 29th, 2012, 01:41 PM  #4 
Global Moderator Joined: May 2007 Posts: 6,641 Thanks: 625  Re: complex number calculation
Every value of n gives a different angle. Proof by contradiction: Assume two values (n and m) give the same angle, then xm  xn would be an integer k, so x = k/(nm) which is rational. 

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