Euler’s Identity

Due to a math problem set, I have absolutely no time to write a lengthy post today. But I’ll share what really rattled my mind in high school:

e^{i \pi} + 1 = 0\,\!

My first thought was that this identity doesn’t make sense. How can you take e to an imaginary power?

Anyways, the fun part is staring at it and appreciating all the elements involved here in their elegant simplicity:

i – an imaginary number

1 – the identity in multiplication

0 – the identity in addition

pi – the commonly used irrational number

e – the commonly used transcendental number

Now the not-so-fun part is proving it (from the power series definition of e^x). Good luck with that.


P.S. Actually both pi and e are transcendental.



Posted on November 24, 2010, in Math and tagged . Bookmark the permalink. Leave a comment.

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