# Definition:Cartesian Product/Cartesian Space/Two Dimensions

## Definition

Let $S$ be a set.

$S^2 = S \times S = \set {\tuple {x_1, x_2}: x_1, x_2 \in S}$

The set $S^2$ called a cartesian space of $2$ dimensions.

### Cartesian Plane

When $S$ is the set of real numbers $\R$, the cartesian product takes on a special significance.

A 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.

## Source of Name

This entry was named for RenĂ© Descartes.