Cartesian coordinate plane
Cartesian coordinate plane
The Cartesian coordinate system is named after French mathematician and philosopher Ren´ Descartes (1596–1650), who was among the first to describe its properties. However, historical evidence shows that Pierre de Fermat (1601–1665), another French mathematician and scholar, actually did more to develop the Cartesian system.
To best understand the nature of the Cartesian plane, we can start with the number line. Begin with line L, which stands for a number axis (Figure 1). On L choose a point, O, and let this point designate the zero point or origin. Distance to the right of O is positive; to the left is negative. Now choose another point, A, to the right of O on L. This point corresponds to the number 1. The distance between O and A is the unit with which we can locate B, C, D…, corresponding to 2, 3, 4. . . Repeat this process to the left of O on L with points Q, R, S, T. . . which 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 (Figure 1). If we further subdivide the segment OA into d equal parts, then 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 can find points that correspond to some irrational numbers as well. For example, in Figure 1, rectangle OCPZ has a base of 3 and a segment of 2. Using the Pythagorean theory we know that the segment OP has a length equal to . Similarly, the length of segment OW is . 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? In the rectangular Cartesian plane, a point’s position is determined with reference to two perpendicular lines called coordinate axes. The intersection of these axes is called the origin, and the four sections into which the axis divide a given plane are 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 (Figure 2). Distance along the x axis to the right of the y axis is positive; to the left is negative. Distance along the y axis above the x axis is positive; below is negative. Ordinarily, the unit of measure along the coordinate axes is the same for both axes, but sometimes it is convenient to use different measures for each axis.
P1 (x1, y1) denotes a fixed point where x1 represents the x coordinate (abscissa)—the perpendicular distance from the y axis to P1 ; y1 represents the y coordinate (ordinate)—the perpendicular distance
from the axis to P1 . For P1, 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 that 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.