Complex Numbers are Uncountable

Theorem

The set of complex numbers $\C$ is uncountably infinite.

Proof

For all $r \in \R$, we have $r = r + 0 i \in C$.

Thus the set of real numbers $\R$ can be considered a subset of $\C$.

As the Real Numbers are Uncountable, it follows from Sufficient Conditions for Uncountability, proposition $(1)$, that $\C$ is uncountably infinite.

$\blacksquare$