Definition:Cartesian Plane

Definition


Every point in the plane can be identified uniquely by means of an ordered pair $\tuple {x, y}$ of real coordinates.

Two perpendicular straight lines are chosen.

These are called the axes.

They are understood to be infinite.

The usual directions to make them are:
 * $(1): \quad$ Across the page, from left to right. This is usually called the $x$-axis.
 * $(2): \quad$ Up the page, from bottom to top. This is usually called the $y$-axis.

The point of intersection of the axes is called the origin.

A unit length is specified.

The axes are each identified with the set of real numbers $\R$, where the origin is identified with $0$.

The real numbers increase to the right on the $x$-axis, upwards on the $y$-axis.

Thus:
 * to the left of the origin the numbers on the $x$-axis are negative
 * below the origin the numbers on the $y$-axis are likewise negative.

Thus the plane can be identified with the cartesian product $\R^2$.

In this context, $\R^2$ is called the (cartesian) coordinate plane.

Also known as
The cartesian coordinate plane is often seen referred to as the $x y$-plane, or (without the hyphen) the $x y$ plane.

Some sources refer to it as the Euclidean plane, but on that term is reserved for the abstract geometry consisting of $\R^2$ together with the set of straight lines.