Euler's Identity

One of the simplest and most elegant equations is Euler's Identity:
e^πi + 1 = 0.

It states that Euler's number e (which is equal to roughly 2.718...) to the power of i*π (taken at roughly 3.14...) is equal to -1. This equation is used to derive cyclotomic integers which are used in Kummer's proof for Fermat's Last Theorem for regular primes.

To be fair, for many people, xⁱ does not have a clear value. What does it mean to put an exponent to an imaginary power? In mathematics, it is ok if the details are not intuitive as long as they are logically consistent. The Maclaurin Series, for example, can be used to define exponents to a complex power (see here). Newton did something similar when he generalized the Binomial Theorem to include complex powers (see here for details)

Theorem: Euler's Identity is the equation: e^πi = -1

This equation derives directly from Euler's Formula:
e^ix = cos x - isin x [See here for proof]

We get:
e^πi = cos(π) + isin(π) = -1 + i(0) = -1. [Review of e (see here), sin and cos (see here), i (see here), and π (see here)]

QED

Corollary: e^πi + 1 = 0

This directly follows from above.

QED