ThmDex – An index of mathematical definitions, results, and conjectures.

Result R424 on D786: Standard natural complex exponential function

Euler's identity

Formulation 0

Let $i$ be the D371: Imaginary number.
Let $\pi$ be the D168: Pi.

Then \begin{equation} e^{i \pi} + 1 = 0 \end{equation}

Proofs

Proof 0

Let $i$ be the D371: Imaginary number.
Let $\pi$ be the D168: Pi.

Applying R425: Euler's formulas for a real variable, we have \begin{equation} \begin{split} e^{i \pi} + 1 & = \cos \pi + i \sin \pi + 1 \\ & = - 1 + i \cdot 0 + 1 \\ & = - 1 + 1 \\ & = 0 \end{split} \end{equation} $\square$