Cardinality of a line and a half plane intuitively it seems like the cardinality of the set of points that make up a line should be different than the cardinality of the set of points that make up a half plane but I couldn't come up with a proof, does a simple proof exist?

It is easiest to start with showing that the unit interval and unit square have the same cardinality.  Ignoring the issue of decimals that end in an infinite series of $9$s (which can be patched up), take $x \in (0,1)$ and $(y,z) \in (0,1) \times (0,1)$  Construct a bijecton by taking all the odd place digits of $x$ to make $y$ and the even place digits of $x$ to make $z$.  So $0.123456789\dots \Longleftrightarrow (0.13579\dots, 0.2468\dots)$Now use your favorite bijection between the interval and the (half) line three times and you are done.
