Helen Lewis

2022-01-16

Why does ${e}^{\frac{i\pi }{2}}=i?$

Donald Cheek

Expert

${e}^{i\theta }\equiv \mathrm{cos}\theta +i\mathrm{sin}\theta$
This is an IDENTITY, meaning for whatever value of $\theta .LHS=RHS$.
Using $\theta =\frac{1}{2}\pi$,
$LHS={e}^{i\frac{1}{2}\pi }$
$RHS=\mathrm{cos}\left(\frac{1}{2}\pi \right)+i\mathrm{sin}\left(\frac{1}{2}\pi \right)$
$=0+i\left(1\right)$
$=i$

Ronnie Schechter

Expert

alenahelenash

Expert

${e}^{\frac{ix}{2}}=i⇔\sqrt{{e}^{i\pi }}=i⇔{\sqrt{{e}^{i\pi }}}^{2}={i}^{2}⇔{e}^{i\pi }=-1⇔{e}^{i\pi }+1=0$ which is Euler's identity