Definition:Cartesian Coordinate System

2 Dimensions


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:
 * Across the page, from left to right. This is usually called the $x$-axis.
 * 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.

Identification of Point in Plane with Ordered Pair
Any point on the plane is now able to be identified by means of a pair of coordinates $\left({x, y}\right)$, as follows:

Select a point $P$ on the plane different from the origin $O$.

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

Draw a line through $O$ and $P$ and call it the $X$-axis.

Draw a 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 distance of the line segment from $Q$ to the $Y$-axis is known as the $X$ coordinate


 * The distance of the line segment from $Q$ to the $X$-axis is known as the $Y$ coordinate.

Thus the $Y$ coordinate of all points on the $X$-axis is zero, and the $X$ coordinate of all points on the $Y$ axis is zero.

The point $P$ is identified with the coordinates $\left({1, 0}\right)$.

Coordinate Plane
The plane can therefore be identified with the cartesian product $\R^2$.

In this context, $\R^2$ is called the (cartesian) coordinate plane.

Quadrants
For ease of reference, the coordinate plane is often divided into four regions by the axes:
 * The area above the $x$-axis and to the right of the $y$-axis is called the first quadrant
 * The area above the $x$-axis and to the left of the $y$-axis is called the second quadrant
 * The area below the $x$-axis and to the left of the $y$-axis is called the third quadrant
 * The area below the $x$-axis and to the right of the $y$-axis is called the fourth quadrant

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

However, it is likely that Descartes was himself influenced significantly by the writings of Nicole Oresme, who had a similar idea some three centuries earlier.