Euler's identity

In mathematics, Euler's identity (also known as Euler's equation) is the equality

where

Euler's identity is named after the Swiss mathematician Leonhard Euler. It is considered to be an example of mathematical beauty.

Euler's identity is a special case of Euler's formula from complex analysis, which states that for any real number x,

where the inputs of the trigonometric functions sine and cosine are given in radians.

In particular, when x = π, or one half-turn (180°) around a circle:

Since

and

it follows that

which yields Euler's identity:

Euler's identity is often cited as an example of deep mathematical beauty. Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:

Furthermore, the equation is given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.

Stanford University mathematics professor Keith Devlin has said, "like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence". And Paul Nahin, a professor emeritus at the University of New Hampshire, who has written a book dedicated to Euler's formula and its applications in Fourier analysis, describes Euler's identity as being "of exquisite beauty".

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