Definition:Real Number Plane

Theorem
The points on the plane are in one-to-one correspondence with the $\R$-vector space $\R^2$.

So from the definition of an ordered $n$-tuple, the general element of $\R^2$ can be defined as an ordered couple $\left({x_1, x_2}\right)$ where $x_1, x_2 \in \R$, or, conventionally, $\left({x, y}\right)$.

Thus, we can identify the elements of $\R^2$ with points in the plane and refer to the point as its coordinates.

Thus we can refer to $\R^2$ as the plane.

Proof
This is shown in Ordered Basis for Coordinate Plane.