# Euler's Identity

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

Euler's identity, , has been called the most beautiful equation in mathematics. It unites the most basic numbers of mathematics: (the base of the natural logarithm), (the imaginary unit = ), (the ratio of the circumference of a circle to its diameter), 1 (the multiplicative identity), and 0 (the additive identity) with the basic arithmetic operations: addition, multiplication and exponentiation—using each once and only once! The identity is a particular example of the more general Euler formula: . The series expansion of is with partial sums .

[more]
Contributed by: Jim Kaiser (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

Euler recognized that the Taylor series expansion of when applied to an imaginary number also converged and, in fact, converged to . The terms of the series for a pure imaginary number alternate between real and imaginary numbers. This provides a very interesting visualization of the convergence of the series. There are many sources for Euler's work with this formula, but [1] is the one that motivated me to create this Demonstration.

Reference

[1] R. P. Crease, *The Great Equations*, New York: W. W. Norton & Company, 2008 pp. 91–106.

## Permanent Citation