Cartesian axes and representation of points in the plane

A Cartesian axes are a pair of perpendicular real lines that allow us to identify the different points in the plane. We will identify any point $P$ by means of a pair of numbers, $a$ and $b$, and we will write $P=(a,b)$. Before explaining how to find $a$ and $b$, let's analyze the Cartesian axes more in depth.

This is a graphic representation of the Cartesian axes:

We observe that we have two real lines that cross at point $0$ of both.

It is woth mentioning that the above mentioned straight lines divide the plane in four parts called quadrants, which are identified accordingly to the figure:

The different axes have their own names:

• The horizontal axis is the abscissa axis.
• The vertical axis is the axis of ordinates.

The point where the two axes are cut is called the origin (sometimes simply $O$), and it takes as its coordinates $O=(0,0)$.

Once the notation has been seen, we are already able to locate points.

A rigorous definition of what is considered to be coordinates of a point might be:

Given a Cartesian axes and a point $P$ of the plane, if $a$ and $b$ are the values of the projection of the point $P$ on the abscissa and ordinates axes, respectively, then we have $P=(a,b)$.

A more constructive definition might be the following one:

The coordinates $a$ and $b$ of a point $P$ of the plane, $P=(a,b)$, are the points of intersection of the parallel lines to the axes of coordinates drawn from point $P$ to the coordinates axes. The first coordinate $a$ is the intersection with the horizontal axis or the abscissa axis, and the second coordinate $b$ is the intersection with the vertical axis or the ordinate axis.

A visual example will turn out to be a lot more clearer.