# Definition:Cartesian Product/Cartesian Space/Two Dimensions

## Definition

Let $S$ be a set.

The **cartesian $2$nd power of $S$** is:

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

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.

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:

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:

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.

## Also see

## Source of Name

This entry was named for RenĂ© Descartes.

## Sources

- 1964: W.E. Deskins:
*Abstract Algebra*... (previous) ... (next): $\S 1.2$: Definition $1.4$ - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 1.7$. Pairs. Product of sets: Example $22$ - 1972: A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*... (previous) ... (next): $\S 2$: Sets and functions: Graphs and functions - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets