# Definition:Cartesian Coordinate System/Coordinate Plane

## Contents

## Definition

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 $\tuple {1, 0}$.

## Also known as

The **cartesian coordinate plane** is often seen referred to as the **$xy$-plane**, or (without the hyphen) the **$xy$ plane**.

## Historical Note

The Cartesian plane was supposedly invented by René Descartes.

However, a study of the literature will reveal that this idea originated considerably earlier, perhaps going back as far as Nicole Oresme.

## Sources

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{III}$: Gentleman, Soldier and Mathematician - 1958: P.J. Hilton:
*Differential Calculus*... (next): Chapter $1$: Introduction to Coordinate Geometry - 1959: E.M. Patterson:
*Topology*(2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Topological Spaces: $\S 8$. Notations and definitions of set theory - 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 2$. Geometrical Representations - 1964: William K. Smith:
*Limits and Continuity*... (previous) ... (next): $\S 2.1$: Sets: Exercise $\text{C} \ 6$ - 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 1.7$. Pairs. Product of sets: Example $23$ - 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 5$: Trigonometric Functions - 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $8$: Field Extensions: $\S 40$. Construction with Ruler and Compasses - 1971: Robert H. Kasriel:
*Undergraduate Topology*... (previous) ... (next): $\S 1.9$: Cartesian Product - 1972: Murray R. Spiegel and R.W. Boxer:
*Theory and Problems of Statistics*(SI ed.) ... (previous) ... (next): Chapter $1$: Rectangular co-ordinates - 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*... (previous) ... (next): $\S 3$. Ordered pairs; cartesian product sets - 1975: Bert Mendelson:
*Introduction to Topology*(3rd ed.) ... (previous) ... (next): Chapter $1$: Theory of Sets: $\S 5$: Products of Sets - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 8$: Cartesian product of sets - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.2$: Sets: Example $\text{(iii)}$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Cartesian coordinate system** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Cartesian coordinate system** - 2010: Raymond M. Smullyan and Melvin Fitting:
*Set Theory and the Continuum Problem*(revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 7$ Cartesian products: Remark