Euler Formula

The Euler formula, sometimes also called the Euler identity, states

$\displaystyle e^{ix}=cosx+isinx,$

where ${i}$ is the imaginary unit. Note that Euler’s polyhedral formula is sometimes also called the Euler formula, as is the Euler curvature formula.

The special case of the formula with ${x=\pi}$ gives the beautiful identity

$\displaystyle e^{i\pi}=-1,$

an equation connecting the fundamental numbers ${i,\pi,e,1,}$ and ${0}$, the fundamental operations ${+}$, ${\times}$, and exponentiation, the most important relation ${=}$, and nothing else. Gauss is reported to have commented that if this formula was not immediately obvious, the reader would never be a first-class mathematician.