For any complex number $z$ define $P ( z ) =$ the cardinality of $\left\{ z ^ { k } \mid k \text{ is a positive integer} \right\}$, i.e., the number of distinct positive integer powers of $z$. It may be useful to remember that $\pi$ is an irrational number. (a) For each positive integer $n$ there is a complex number $z$ such that $P ( z ) = n$. (b) There is a unique complex number $z$ such that $P ( z ) = 3$. (c) If $| z | \neq 1$, then $P ( z )$ is infinite. (d) $P \left( e ^ { i } \right)$ is infinite.
For any complex number $z$ define $P ( z ) =$ the cardinality of $\left\{ z ^ { k } \mid k \text{ is a positive integer} \right\}$, i.e., the number of distinct positive integer powers of $z$. It may be useful to remember that $\pi$ is an irrational number.
(a) For each positive integer $n$ there is a complex number $z$ such that $P ( z ) = n$.\\
(b) There is a unique complex number $z$ such that $P ( z ) = 3$.\\
(c) If $| z | \neq 1$, then $P ( z )$ is infinite.\\
(d) $P \left( e ^ { i } \right)$ is infinite.