Cartesian Coordinate System

We saw that a point P on a number line can be specified by a real number $$x$$ called its coordinate. Similarly, by using a Cartesian coordinate system named in honor of the French philosopher and mathematician René Descartes (1596—1650), we can specify a point P in the plane with two real numbers, also called coordinates. $$\left( {x,y} \right)$$

A Cartesian coordinate system consists of two perpendicular number lines, called coordinate axes, which meet at a common origin $$O$$, as shown in the figure below. Ordinarily, one of the number lines, called the $$x$$ axis, is horizontal, and the other, called the y axis, is vertical. Numerical coordinates increase to the right along the $$x$$ axis, and upward along the $$y$$ axis. We usually use the same scale (that is, the same unit distance) on the two axes, although in some of our figures, space considerations make it convenient to use different scales.

If $$P$$ is a point in the plane, the coordinates of $$P$$ are the coordinates $$x$$ and $$y$$ of the points where perpendiculars from $$P$$ meet the two axes, as shown in the figures. The $$x$$ coordinate is called the abscissa of $$P$$, and the y coordinate is called the ordinate of $$P$$. The coordinates of $$P$$ are traditionally written as an ordered pair $$\left( {x,y} \right)$$ enclosed in parentheses, with the abscissa first and the ordinate second.

To plot the point $$P$$ with coordinates $$\left( {x,y} \right)$$ means to draw Cartesian coordinate
axes and to place a dot representing $$P$$ at the point with abscissa $$x$$ and ordinate $$y$$. You can think of the ordered pair $$\left( {x,y} \right)$$ as the numerical “address” of  $$P$$. The correspondence between $$P$$ and $$\left( {x,y} \right)$$ seems so natural that in practice we identify the point $$P$$ by its “address” $$\left( {x,y} \right)$$ by writing $$P = \left( {x,y} \right)$$. With this identification in mind, we call an ordered pair of real numbers $$\left( {x,y} \right)$$ a point, and we refer to the set of all such ordered pairs as the Cartesian plane or the $$xy$$ plane.

The $$x$$ and $$y$$ axes divide the plane into four regions called quadrants I, II, III and IV as shown in the figures above. Quadrant I consists of all points $$\left( {x,y} \right)$$ for which both $$x$$ and $$y$$ are positive, quadrant II consists of all points $$\left( {x,y} \right)$$ for which $$x$$ is negative and $$y$$ is positive, and so forth. Notice that a point on a coordinate axis belongs to no quadrant.