Definition:Cartesian Coordinate System

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:

A generic point in the Cartesian Coordinate 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.

A unit length is specified.

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 $\tuple {1, 0}$.

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

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.

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.

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.

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.