Complex numbers

(cos(x)+isin(x))^n=cos(nx)+isin(nx)

Now using maclaurine series

e^ix=1+x+x^2/2!+x^3/3!+....+x^r/r!+...

this can be used to that

e^ix=cos(x)+isin(x)

which means

e^inx=cos(nx)+isin(nx)

this is then used to show that when n=1 x=pi

e^(ipi)=-1

e^(ipi)+1=0

This result has been describe by many as the most beautiful equation in Maths.