Cartesian plane

René DesCartes

In the 16th century the French mathematician and philosopher René DesCartes was one of many mathematicians working on how to integrate geometry, which western civilization had inherited from the ancient Greeks, with the algebra that had been brought to Europe from Islamic civilization in the 11th century. He introduced a way of representing the geometry of algebraic relationships—what we call equations—by plotting the values of two variables together on a rectangular grid, with distances in the horizontal direction representing changes in the value of the first variable, and distances in the vertical direction representing changes in the value of the second variable. We call this grid the Cartesian plane or the Cartesian coordinate system.

Figure 1: The Cartesian plane.

The range of values of the first variable are indicated by a horizontal axis, those of the second variable by a vertical axis, and these axes intersect at the point where both are zero, at what is called the origin. Most often in algebra or calculus these are labeled the \(x\)-axis and the \(y\)-axis respectively, but they are in any event labeled with the symbols of the respective variables, whatever they may be.

By convention the four quadrants into which the plane is divided by the axes are referred to as quadrants I through IV, with quadrant I being the quadrant in which the values of both variables are positive, and the other quadrants numbered anti-clockwise. It is not required that the aspect ratio between the axes be the same. Rather, since the purpose of the Cartesian plane is to allow useful visual representations of numerical relations, the axes should be marked and the origin placed in whatever way promotes this.

A point in the Cartesian plane is referred to as a coordinate pair or coordinates, with the two values separated by a comma and enclosed by parentheses or angle brackets.

Figure 2

The first value in the coordinate pair is called the abscissa and the second is called the ordinate.

Developing a facility with drawing graphs of functions and relations in the Cartesian plane is one of the most important stages in the learning of mathematics.