De Moivre's Theorem

If we repeat the process of the above section over and over again we can show that if z $\neq$ 0 and n is a positive whole number then | zⁿ| = | z|ⁿ and n arg(z) is an argument of zⁿ.

Since $$\left\vert\vphantom{\frac{1}{z}}\right.$$$${\frac{{1}}{{z}}}$$$$\left.\vphantom{\frac{1}{z}}\right\vert$$ = $${\frac{{1}}{{\vert z\vert}}}$$ and arg(1/z) = - arg(z) we also get the same results if n is a negative integer. So we have the above formulae for all integer values of n (n = 0 is easy -- check).

The result is often put in the following useful form, which is known as de Moivre's Theorem. If n is any whole number then

$$\boxed{z = r(\cos\theta + j\sin\theta) \quad\Rightarrow\quad z^n = r^n(\cos n\theta + j\sin n\theta)}$$

Let me tell you of one other notation at this point, which looks a bit obscure at the moment but which you will meet a lot in later years:

$$\boxed{e^{j\theta} = \cos\theta + j\sin\theta}$$

So

z = r(cos$$\theta$$ + j sin$$\theta$$) = re^j$\theta$    and    zⁿ = rⁿe^nj$\theta$.