Euler's Formula

Euler's formula is the following identity in complex analysis making use of the imaginary number $i$:

$e^{i\phi }=\cos \phi +i\sin \phi \!$

Euler's Identity

The formula can be used to derive an equation known as Euler's identity:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e^{i\pi} + 1 = 0 \!}

This is called the most beautiful equation in all of mathematics, because it equates the five most common mathematical constants in a perfectly elegant manner.

See also

• De Moivre's Theorem
• Leonard Euler
• Arbitrariness or Ambiguity in the Definition of the Imaginary Unit in Mathematics