Definition:Real Number Plane

Theorem
The points on an infinite flat plane are in one-to-one correspondence with the $\mathbb{R}$-vector space $$\mathbb{R}^2$$.

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

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

Thus we can refer to $$\mathbb{R}^2$$ as "the plane".