Below is Vi Hart's video explaining why τ (= 2π) is the more natural "circle constant". Everything just works out more neatly with τ and thinking about a full turn rather than a half turn just makes sense. And yes,
eiτ = 1is how it should be. The Tau Manifesto has more details but suffice to say, while Mar 14th never did much for me, on Jun 28th I might just make an effort this year.
No this is not entirely a waste of time, using τ with beginners would actually eliminate quite a bit of confusion.