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 $e^{\pi\sqrt{-1}}$?

  • 1
    $\begingroup$ Maybe it's hard-coded? $\endgroup$ Dec 31, 2015 at 5:36
  • $\begingroup$ Maybe $e^{\phi\sqrt{-1}}$ is hardcoded as $\operatorname{cis}\phi$ $\endgroup$ Dec 31, 2015 at 5:37
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    $\begingroup$ Maybe it can't display complex numbers? $\sqrt{-1}$ is complex but $e^{\pi\sqrt{-1}}$ isn't. (I really doubt this though - it'd be impressive for a calculator to determine that $e^{\pi\sqrt{-1}})$ really has a $0$ imaginary part and not just a $10^{-99}$ imaginary part or something) $\endgroup$ Dec 31, 2015 at 5:38
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    $\begingroup$ Can it "do" $e^{0.5\pi\sqrt{-1}}$? $\endgroup$
    – MPW
    Dec 31, 2015 at 5:40
  • $\begingroup$ @MPW No, returns error. $\endgroup$
    – ZTqvhI5vpo
    Dec 31, 2015 at 5:42

1 Answer 1


$e^{\pi\sqrt{-1}}=\cos \pi + \sqrt{-1}\sin \pi=-1+0=-1$ which is a real number


$\sqrt{-1}=i$ 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.

  • $\begingroup$ @MiloBrandt Thanks. Typo corrected. :-) $\endgroup$ Dec 31, 2015 at 5:40
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    $\begingroup$ Interesting. So you are saying it can do complex calculations, but not display them? That seems like a waste. How hard could it be to display complex values? $\endgroup$
    – ZTqvhI5vpo
    Dec 31, 2015 at 5:43
  • $\begingroup$ @JamieSanborn I have a CASIO fx-3650 SUPER-FX calculator which can show complex numbers and make different calculations like argument,modulus, etc. But it still cannot display the entire complex number in one screen at a time. It first displays the real part and then you need to toggle the screen to see the imaginary part with a small "i" visible in the right corner. $\endgroup$ Dec 31, 2015 at 5:47
  • $\begingroup$ What do you mean by "Computation of complex numbers is possible in any calculator"? $\endgroup$
    – Nex
    Dec 31, 2015 at 5:54
  • $\begingroup$ @Nex It can calculate $\sqrt{-1}$ but then the calculator realises it is an imaginary number and hence fails to display the result due to absence of proper hardware and settings. $\endgroup$ Dec 31, 2015 at 5:57

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