This might be a simple question, but are there "more" points on a line in (3D) space than on a plane? Or in more mathematical terms: Do $\mathbb {R}$ and $\mathbb {R}^2$ have equal cardinalities (which $\mathbb {N}$ and $\mathbb {N}^2$ have).

If this is true, what could a bijection/pairing function ($\pi \colon\ \, \mathbb {R}^2 \to \mathbb {R}$) for reals look like?