The real numbers in the open unit interval  $I\,=\,(0,\,1)$  can be uniquely represented by their decimal expansions, when these must not end in an infinite string of 9’s.  Correspondingly, the elements of the open unit square $I\!\times\!I$ are represented by the pairs of such decimal expansions.

Let

be such a pair representing an arbitrary point in $I\!\times\!I$ and let

Then it’s apparent that

is an injective mapping from $I\!\times\!I$ to $I$.  Thus

But since $I\!\times\!I$ contains more than one horizontal open segment equally long as $I$ (and accordingly there is a natural injection from $I$ to $I\!\times\!I$), we must have also

The conclusion is that

i.e. that the sets $I\!\times\!I$ and $I$ have equal cardinalities, and the Schröder–Bernstein theorem even garantees a bijection between the sets.

Remark.  Georg Cantor utilised continued fractions for constructing such a bijection between the unit interval and the unit square; cf. e.g. this MAA article.

Since the mapping  $g\!:\,I\to\mathbb{R}$  defined by

is bijective, we can conclude that the sets $\mathbb{R}$ and $\mathbb{R}\!\times\!\mathbb{R}$, i.e. the set of the points of a line and the set of the points of a plane, have the same cardinalities.  This common cardinality is $2^{{\aleph_{0}}}$.