Properties of the arguments

1.the argument of the product of two complex numbers is equal to the sum of their arguments.
i.e. arg(z1z2)=argz1 + argz2
proof:
let z1=r1e^i?1 , z2=r2e^i?2
? z1 z2=r1 r2 e^i?1 e^i?2= r1 r2 e^(i(?1+?2))= r1 r2 (cos(?1+?2)+isin( ?1+?2)
? arg(z1 z2)=argz1+argz2
2.the argument of the quotient of two complex numbers is equal to the different of their arguments.
i.e. arg(z1/z2)=argz1-argz2
proof:
let z1=r1e^i?1 , z2=r2e^i?2
z1 / z2=(r1 e^i?1)/(r2e^i?2 )=r1/r2 e^(i(?1-?2))=r1/r2(cos(?1-?2)+isin(?1-?2))
? arg(z1/z2)=argz1-argz2
6.products and quotients in exponential form:
if z1=r1e^i?1 , z2=r2e^i?2
? z1 z2=r1 r2 e^i?1 e^i?2= r1 r2 e^(i(?1+?2))................(1)
Moreover
z1 / z2=(r1 e^i?1)/(r2e^i?2 ). ( e^(-i?2))/e^(-i?2) =(r1 )/r2 e^(i(?1-?2)).................(2)
because
e^0i=1
Thus
1/z=z-1=(1 )/r e^(-i?)…………………………………………(3)
Another important result that can be obtained formally by applying rules for real number is
Zn=rne^in? (n=0,?1,?2,……………..)
=rn(cosn?+isinn?)
So that ,we get de Moiver’s formula
(cos?+isin?)n= cosn?+isinn?………….(5)