When I type in the identity $e^{\pi\sqrt{-1}}$ on my phone calculator (LG phone running Android), I get the correct result of $-1$
However, when I simply type $\sqrt{-1}$, it returns an error.
Why can the calculator do $e^{\pi\sqrt{-1}}$, but not do $\sqrt{-1}$ if $\sqrt{-1}$ is a direct part of $e^{\pi\sqrt{-1}}$?
Best Answer
$e^{\pi\sqrt{-1}}=\cos \pi + \sqrt{-1}\sin \pi=-1+0=-1$ which is a real number
BUT
$\sqrt{-1}=i$ is a complex number with a zero real part and a non-zero imaginary part.
Computation of complex numbers is possible in any calculator but showing the results containing imaginary numbers is not possible except in certain high-grade calculators.