A) a purely real number
B) a purely complex number
C) a complex number whose real part is always a negative real number
D) a complex number whose real part is always a positive integer
Correct Answer: A
Solution :
Let \[(a+ib)={{i}^{i}}\] Taking log on both sides, we get \[\log (a+ib)=i\,\log \,i\] \[\Rightarrow \] \[\log (a+ib)=i\left( i\frac{\pi }{2} \right)\] \[\Rightarrow \] \[\log (a+ib)=-\frac{\pi }{2}\] \[\Rightarrow \] \[(a+ib)={{e}^{-\pi /2}}\] Hence, \[{{i}^{i}}\] is purely real number.
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