A remarkable identity in mathematics is the following: \[e^{i\pi} + 1 = 0.\] This identity relates five fundamental mathematical constants and is called “the most beautiful math formula” by some.
A proper proof of this identity involves complex analysis and is beyond the scope of these notes. The key to proving the identity is the following more general result: \[e^{ix} = \cos x + i\sin x.\] Then, setting \(x = \pi\) gives us Euler's identity.
Watch Salman Khan's video below for an informal sketch of the proof that \(e^{ix} = \cos x + i\sin x.\)