\begin{definition}[Definition:Cartesian Plane]
324pxrightthumbA general point $Q = \tuple {x, y}$ in the Cartesian plane
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.
\end{definition}