Points of the Unit Circle and the Polar Form of a Nonzero Complex Number

In the setting of Complex Analysis: Standing Notation, let $\exp:\mathbb{C}\to\mathbb{C}$ be the complex exponential function, and let $\pi$ be the real number of The Number Pi (i).

(i) Points of the unit circle. For every $z\in\mathbb{C}$ with $|z|=1$ there is exactly one real number $t$ with $0\le t<2\pi$ and

$$\exp(it)=z .$$

(ii) Polar form. For every $z\in\mathbb{C}$ with $z\neq0$ there is exactly one choice of real $r$ and $t$ with $r>0$ and $0\le t<2\pi$ such that

$$z=r\exp(it),$$

and then $r=|z|$.