Explanation?

The head of the physics department came to us today and said...


e^(pi)i = -1

and

e^(2pi)i = 1

i stands for an imaginary number, can anyone explain this to me ?

If you develop e^iz as a Taylor series around z=0 you get:

e^iz = 1 + iz - z²/2! - iz³/3! + z⁴/4! + iz⁵/5! - z⁶/6! - iz⁷/7! + ... =

(1 - z²/2! + z⁴/4! - z⁶/6! + ...) + i(z - z³/3! + z⁵/5! - z⁷/7! + ...) =

cos(z) + i sin(z)

If z is real this holds only for small values (I seem to recall ABS(z) <=1), but for z complex there's no such limitation (though I can't remember how to prove it).

So e^iz = cos(z) + i sin(z)

and therefore, e^iPi = cos(Pi) + i sin(Pi) = -1

and e^i2Pi = (e^iPi)² = (-1)² = 1