# Definition:Cartesian Coordinate System

## Contents

## Definition

A **Cartesian coordinate system** is a coordinate system in which the position of a point is determined by its relation to a set of coordinate axes.

### Cartesian Plane

In $2$ dimensions, the **Cartesian coordinate system** is referred to as the **cartesian plane**:

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 $\tuple {1, 0}$.

## Quadrants

For ease of reference, the cartesian coordinate plane is often divided into four quadrants by the axes:

### First Quadrant

Quadrant $\text{I}: \quad$ The area above the $x$-axis and to the right of the $y$-axis is called **the first quadrant**.

That is, **the first quadrant** is where both the $x$ coordinate and the $y$ coordinate of a point are positive.

### Second Quadrant

Quadrant $\text{II}: \quad$ The area above the $x$-axis and to the left of the $y$-axis is called **the second quadrant**.

That is, **the second quadrant** is where the $x$ coordinate of a point is negative and the $y$ coordinate of a point is positive.

### Third Quadrant

Quadrant $\text{III}: \quad$ The area below the $x$-axis and to the left of the $y$-axis is called **the third quadrant**.

That is, **the third quadrant** is where both the $x$ coordinate and the $y$ coordinate of a point are negative.

### Fourth Quadrant

Quadrant $\text{IV}: \quad$ The area below the $x$-axis and to the right of the $y$-axis is called **the fourth quadrant**.

That is, **the fourth quadrant** is where the $x$ coordinate of a point is positive and the $y$ coordinate of a point is negative.

Note that the axes themselves are generally not considered to belong to any quadrant.

## Source of Name

This entry was named for René Descartes.

## 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.

## Also known as

**Cartesian coordinates** are also known as **rectangular coordinates**.

## Also see

- Results about
**cartesian coordinate systems**can be found here.

## Sources

- 1936: Richard Courant:
*Differential and Integral Calculus: Volume $\text { II }$*... (next): Chapter $\text I$: Preliminary Remarks on Analytical Geometry and Vector Analysis: $1$. Rectangular Co-ordinates and Vectors: $1$. Coordinate Axes - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Graphical Representation of Complex Numbers - 1981: Murray R. Spiegel:
*Theory and Problems of Complex Variables*(SI ed.) ... (previous) ... (next): $2$: Curvilinear Coordinates - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Cartesian coordinate system** - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $2$: 'And you do addition?': $\S 2.1$: Functions - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Cartesian coordinate system** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $6$: Curves and Coordinates: Descartes