In analytical mathematics, Euler’s identity (also known as Euler’s equation), named for the Swiss-German mathematician Leonhard Euler, is the equality shown above, where
- e is Euler’s number, the base of natural logarithms,
- π is pi, the ratio of the circumference of a circle to its diameter, and
- i is the imaginary unit, which satisfies i² = −1.
Euler’s identity is considered by many to be remarkable for its mathematical beauty. These three basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:
- The number 0, the additive identity.
- The number 1, the multiplicative identity.
- The number π, which is ubiquitous in trigonometry, the geometry of Euclidean space, and analytical mathematics (π = 3.14159265…)
- The number e, the base of natural logarithms, which occurs widely in mathematical and scientific analysis (e = 2.718281828…). Both π and e are transcendental numbers.
- The number i, the imaginary unit of the complex numbers, a field of numbers that contains the roots of all polynomials (that are not constants), and whose study leads to deeper insights into many areas of algebra and calculus, such as integration in calculus.
Furthermore, in algebra and other areas of mathematics, equations are commonly written with zero on one side of the equals sign.