Cartesian Coordinate Plane

The Cartesian coordinate system is named after René Descartes (1596–1650), the noted French mathematician and philosopher, who was among the first to describe its properties. However, historical evidence shows that Pierre de Fermat (1601-1665), also a French mathematician and scholar, did more to develop the Cartesian system than did Descartes.

To best understand the nature of the Cartesian plane, it is desirable to start with the number line. Begin with line L and let L stand for a number axis (see Figure 1). On L choose a point, O, and let this point designate the zeropoint or origin. Let the distance to the right of O be considered as positive; to the left as negative. Now choose another point, A, to the right of O on L. Let this point correspond to the number 1. We can use this distance between O and A to serve as a unit with which we can locate B, C, D,... to correspond to the +1, +2, +3, +4,... Now we repeat this process to the left of O on L and call the points Q, R, S, T,... which can correspond to the numbers -1, -2, -3, -4,... Thus the points A, B, C, D,..., Q, R, S, T,... correspond to the set of the integers (see Figure 1). If we further subdivide the segment OA into d equal parts, the 1/d represents the length of each part. Also, if c is a positive integer, then c/d represents the length of c of these parts. In this way we can locate points to correspond to rational numbers between 0 and 1.

By constructing rectangles with their bases on the number line we are able to find points that correspond to some irrational numbers. For example, in Figure 1, rectangle OCPZ has a base of 3 and a segment of 2. Using the Theory of Pythagorus we know that the segment OP has a length equal to √13 . Similarly, the length of segment OW is √10 . The real numbers have the following property: to every real number there corresponds one and only one point on the number axis; and conversely, to every point on the number axis there corresponds one and only one real number.

What happens when two number lines, one horizontal and the other vertical, are introduced into the plane? In the rectangular Cartesian plane, the position of a point is determined with reference to two perpendicular line called coordinate axes. The intersection of Figure 1. Illustration by Hans & Cassidy. Courtesy of Gale Group. Figure 2. Cartesian Plane Illustration by Hans & Cassidy. Courtesy of Gale Group.


Figure 3. A plot of the points P ₁(-1, 2); P ₂(3/2,3/4); P ₃(4,0); P ₄(-2, -1). Illustration by Hans & Cassidy. Courtesy of Gale Group. Figure 4. A graph of the triangle whose vertices are the points P ₁(-2,-1), P ₂(6,1), and P ₃(2,4). Illustration by Hans & Cassidy. Courtesy of Gale Group.


these axes is called the origin, and the four sections into which the axis divide a given plane are called quadrants. The vertical axis is real numbers usually referred to as the y axis or functions axis; the horizontal axis is usually known as the x axis or axis of the independent variable. The direction to the right of the y axis along the x axis is taken as positive; to the left is taken as negative. The direction above the x axis along the y axis is taken as positive; below as negative. Ordinarily, the unit of measure along the coordinate axes is the same for both axis, but sometimes it is convenient to use different measures for each axis.

The symbol P₁ (x₁, y₁) is used to denote the fixed point P₁. Here x₁ represents the x coordinate (abscissa) and is the perpendicular distance from the y axis to P₁; y₁ represents the y coordinate (ordinate) and is the perpendicular distance from the axis to P₁. In the symbol P₁ (x₁,y₁), x₁ and y₁ are real numbers. No other kind of numbers would have meaning here. Thus, we observe that by means of a rectangular coordinate system we can show the correspondence between pairs of real numbers and points in a plane. For each pair of real numbers (x, y) there corresponds one and only one point (P), and conversely, to each point (P) there corresponds one and only pair of real numbers (x, y). We say there exists a one-to-one correspondence between the points in a plane and the pair of all real numbers.

The introduction of a rectangular coordinate system had many uses, chief of which was the concept of a graph. By the graph of an equation in two variables, say x and y, we mean the collection of all points whose coordinates satisfy the given equation. By the graph of the function f(x) we mean the graph of the equation y= f(x). To plot the graph of an equation we substitute admissible values for one variable and solve for the corresponding values of the other variable. Each such pair of values represents a point which we locate in the coordinate system. When we have located a sufficient number of such points, we join them with a smooth curve.

In general, to draw the graph of an equation we do not depend merely upon the plotted points we have at our disposal. An inspection of the equation itself yields certain properties which are useful in sketching the curve like symmetry, asymptotes, and intercepts.