Tuesday, August 20, 2019

algebra precalculus - Prove $e^{i pi} = -1$





I recently heard that $e^{i \pi} = -1$.


WolframAlpha confirmed this for me, however, I don't see how this works.


Answer



This identity follows from Euler's Theorem, \begin{align} e^{i \theta} = \cos \theta + i \sin \theta, \end{align} which has many proofs. The one that I like the most is the following (sketched). Define $f(\theta) = e^{-i \theta}(\cos \theta + i \sin \theta)$. Use the quotient rule to show that $f^{\prime}(\theta)= 0$, so $f(\theta)$ is constant in $\theta$. Evaluate $f(0)$ to prove that $f(\theta) = f(0)$ everywhere.



Take $\theta = \pi$ for your claim.


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