Euler's Identity
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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
.
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.
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