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Part of a series of articles on
The mathematical constant e
Natural logarithm · Exponential function
Applications in: compound interest · Euler's identity & Euler's formula · halflives & exponential growth/decay
Defining e: proof that e is irrational · representations of e · Lindemann–Weierstrass theorem
People John Napier · Leonhard Euler
Schanuel's conjecture
In analytical mathematics, Euler's Identity, named for the SwissGerman mathematician Leonhard Euler, is the equality
where
Euler's Identity is also sometimes called Euler's Equation.
Contents
Beauty
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.
Full article ▸


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