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Thursday, September 11, 2014

Throwback Fact of the Week - Euler's Identity - 9/11/14

Euler's Identity

Euler's Identity is considered to be one of the most amazing and beautiful mathematical relationships ever discovered. The statement of the equality is as follows:

e^{i \pi} + 1 = 0

You may wonder why this equality is looked upon with such awe and admiration.

The identity seamlessly links some of the most important mathematical symbols: the number 0 (the additive identity), the number 1 (the multiplicative identity), the transcendental numbers π and e (Euler's number), and i (the imaginary unit). If this wasn't enough, it also makes use of three of the basic arithmetic operations: addition, multiplication and exponentiation.

The derivation of the identity follows from Euler's formula, which states:

e^{ix} = \cos x +  i\sin x \,\!

Evaluating this formula at x = π yields the identity. 

Kasner and Newman note, "We can only reproduce the equation and not stop to inquire into its implications. It appeals equally to the mystic, the scientist, and the mathematician."

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