e to the pi times i

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.