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Euler's Identity

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 .
This Demonstration starts with Euler's identity but lets you experiment with the more general formula for various values of . You can see how quickly the series expansion of converges by varying the number of terms in its partial sums with . The green polygonal line joins the points , , …, where . You can zoom in with the scale, which is logarithmic. When you zoom in sufficiently close, the center of the picture switches from the origin to the point being approximated.

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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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