Definition:Cartesian Plane

Definition
The Cartesian plane is a Cartesian coordinate system of $2$ dimensions.

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

Identify one distinct point on the plane as the origin $O$.

Select a point $P$ on the plane different from $O$.

Construct an infinite straight line through $O$ and $P$ and call it the $x$-axis.

Identify the $x$-axis with the real number line such that:
 * $0$ is identified with the origin $O$
 * $1$ is identified with the point $P$

The orientation of the $x$-axis is determined by the relative positions of $O$ and $P$.

It is conventional to locate $P$ to the right of $O$, so as to arrange that:


 * to the right of the origin, the numbers on the $x$-axis are positive
 * to the left of the origin, the numbers on the $x$-axis are negative.

Construct an infinite straight line through $O$ perpendicular to the $x$-axis and call it the $y$-axis.

Identify the point $P'$ on the $y$-axis such that $OP' = OP$.

Identify the $y$-axis with the real number line such that:
 * $0$ is identified with the origin $O$
 * $1$ is identified with the point $P'$

The orientation of the $y$-axis is determined by the position of $P'$ relative to $O$.

It is conventional to locate $P'$ such that, if one were to imagine being positioned at $O$ and facing along the $x$-axis towards $P$, then $P'$ is on the left.

Hence with the conventional orientation of the $x$-axis as horizontal and increasing to the right:


 * going vertically "up" the page or screen from the origin, the numbers on the $y$-axis are positive
 * going vertically "down" the page or screen from the origin, the numbers on the $y$-axis are negative.

Cartesian Coordinate Pair
Hence:
 * the point $P$ is identified with the coordinates $\tuple {1, 0}$
 * the point $P'$ is identified with the coordinates $\tuple {0, 1}$.

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.