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.