Definition:Real Number Plane

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The points on the plane are in one-to-one correspondence with the (real) Cartesian plane $\R^2$.

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

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.

Also see

The validity of this definition is shown in Ordered Basis for Coordinate Plane.