# Definition:Cartesian Product/Cartesian Space/Two Dimensions

## Contents

## Definition

Let $S$ be a set.

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

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

Thus $S^2 = S \times S$

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 points in the plane can be identified uniquely by means of a pair of coordinates.

Two perpendicular straight lines are chosen. These are understood to be infinite. These are called the axes.

The usual directions to make these 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.

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**.

### Identification of Point in Plane with Ordered Pair

Every point on the plane can be identified by means of a pair of 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 $O$.

Let the distance from the origin to $P$ be defined as being $1$.

Draw an infinite straight line through $O$ and $P$ and call it the $X$-axis.

Draw an infinite straight line through $O$ perpendicular to $OP$ and call it the $Y$-axis.

Now, let $Q$ be any point on the plane.

Draw two lines through $Q$, parallel to the $X$-axis and $Y$-axis.

The plane is then conventionally oriented so that the $X$-axis is horizontal with $P$ being to the right of $O$.

Thus the $Y$-axis is then a vertical line.

Thus the point $Q$ can be uniquely identified by the ordered pair $\tuple {x, y}$ as follows:

### X Coordinate

The distance of the line segment from $Q$ to the $Y$-axis is known as the **$X$ coordinate** and (usually) denoted $x$.

If $Q$ is to the right of the $Y$-axis, then $x$ is positive.

If $Q$ is to the left of the $Y$-axis, then $x$ is negative.

The **$X$ coordinate** of all points on the $Y$-axis is zero.

### Y Coordinate

The distance of the line segment from $Q$ to the $X$-axis is known as the **$Y$ coordinate** and (usually) denoted $y$.

If $Q$ is above the $X$-axis, then $y$ is positive.

If $Q$ is below the $X$-axis, then $y$ is negative.

The **$Y$ coordinate** of all points on the $X$-axis is zero.

The point $P$ is identified with the coordinates $\tuple {1, 0}$.

## 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