Cartesian Coordinate System

Not only the world of mathematics, but also many aspects of our daily life have to do in some way with locations in space: the location of a street, the location of our house, the theater seat. Therefore, it is important to know the way in which we can describe the position of a point in space; this is done by means of coordinates.

The coordinate system we use most frequently is the Cartesian (or rectangular) system, named in honor of the famous French philosopher and mathematician, René Descartes (1556 - 1650), who shares with the French lawyer and mathematician Pierre Fermat (1601 - 1665), the credit for his invention.

Cartesian coordinates

Figure i
Figure 2. Parametric equations of the curve

To represent the location of any point P on a plane, we rely on a coordinate system formed by two lines perpendicular to each other, which intersect in a  point zero (Figure I).

  • The lines are called coordinate axes and the point of intersection center or origin.
  • The lines divide the plane into four parts, called quadrants, numbered from I to IV in a counterclockwise direction.
  • The horizontal axis is called x axis or axis of the abscissa and it has its positive part to the right of the center and its negative part to the left.
  • The vertical axis is called axis and or axis of the ordinates and it has its positive part up and its negative part down.
Figure II.
Figure II.

Coordinates of a point

Any point P on the plane is specified by a pair of real numbers called point coordinates. These points are written in order, first that of the horizontal axis (called abscissa) and then the vertical axis (called ordered):

P (abscissa, ordered)

To obtain these coordinates of the point P (x, y), we draw a line parallel to the x axis that passes through P; the cut-off point of this line with the x-axis is at a distance x of origin. Then we draw a line parallel to the axis and through P; the distance of the cut point from the y axis to the origin is y (Figure II).

Conversely, if we know the coordinates of the point and want to locate it on the plane, we first draw a parallel line for its abscissa and then one for its ordinate towards the axes; These lines intersect at a point that is the one represented by the coordinates P (x, y) (Figure III).

Figure III. To determine the point P, it is drawn by the abscissa (-3) and its ordinate (2) lines parallel to the axes; the meeting point of these lines represents point P (-3,2) of the real plane.
Figure III. To determine the point P, it is drawn by the abscissa (-3) and its ordinate (2) lines parallel to the axes; the meeting point of these lines represents point P (-3,2) of the real plane.