e^(pi * i) + 1 = 0

Why? I don't understand this at all ><

a really, really bad proof (but i don't have the time for anything more fancy):
z=cos(x)+i*sin(x)
dz/dx=-sin(x)+i*cos(x)=i*(cos(x)+i*sin(x))=i*z
dz/z=i*dx
ln(z)=ix + c
z=C*exp(ix)

bad one i know