TITLE: Do the real numbers and the complex numbers have the same cardinality?
QUESTION [33 upvotes]: So it's easy to show that the rationals and the integers have the same size, using everyone's favorite spiral-around-the-grid.
Can the approach be extended to say that the set of complex numbers has the same cardinality as the reals?

REPLY [4 votes]: Consult #4b in http://faculty.lasierra.edu/~jvanderw/classes/m415a03/hw8ans.pdf.
A straightforward bijection $B : \mathbb{R}^2 \rightarrow \mathbb{C}$ is: $B(a,b) = a + bi$. I omit the verification of injectivity and surjectivity. Then $|C| = |\mathbb{R}^2|$.
The separate result that $|\mathbb{R}^k| = |\mathbb{R}| \; \forall \; k \in \mathbb{N}$
implies $|\mathbb{R}^2| = |\mathbb{R}|$. Altogether, $|\mathbb{C}| = |\mathbb{R}^2| = |\mathbb{R}|$.