Real Plane
Why
We are constantly thinking of the elements of $\R ^2$ as points of a plane.1
Discussion
We commonly associate elements of $\R ^2$ with points on a plane. (see Geometry).
Given a plane, there exists a set of its
(infinite) lines.
Let $L$ be the set of lines of a plane.
Then $\cup L$ is the set of points of the
plane.
There exists a one-to-one correspondence mapping
elements of $\cup L$ onto elements of $\R ^2$.
For this reason, we sometimes call elements of
$\R ^2$ points.
We call the point associated with $(0, 0)$ the
origin.
We call the element of $\R ^2$ which corresponds
to a point the coordinates
of the point.
Visualization
To visualize the correspondence we draw two perpendicular lines. We then associate a point of the line with $(0, 0) \in \R ^2$. We can label it so. We then pick a unit length. And proceed as usual.2
Given that we have identified a plane with $\R ^2$ in this way, we call $(x, y) \in \R ^2$ the coordinates of the point it corresponds to. Many authors refer to this identification as a Cartesian coordinate system (or Rectangular coordinate system, $x$-$y$ coordinate system).