The Cartesian Coordinate System - Research Paper Example

By logical experience, there is no way Mathematics would have confidently sustained itself apart from at least the faintest notion of coordinates. Set relations as rendered to points that may come in pair up to an infinite number ought to signify the essence both of dependent and independent variables that would be difficult to imagine beyond the two-dimensional rectangular coordinate system in dissociation with each other. The Cartesian Coordinate System does not merely point reference to the graphical means of finding link between variables, rather, it gives Mathematics the desired image of identity in visible shapes and forms by which a learner can gain appreciation of the course as an interesting field of study. The Cartesian Coordinate System Prior to the concept of a two-dimensional system, the discovery of a coordinate system with one dimension had already enabled demonstration of the relative magnitudes of numbers in a graphical manner and had even shown how a distance between two points in the number line may be represented by the magnitude of their differences. The overall advantage, however, of a one-dimensional coordinate system is limited and is unable to extend its applicability to the relation or dependence of two sets of numbers quite significant in the mathematical studies of corresponding values wherein a set constituted by an ordered pair of numbers may be held in association to another or a couple other sets in a planar system of coordinates (Vance, p. 75). Importance of the Cartesian Coordinate System In 1637 Rene Descartes, a French mathematician and philosopher, used the Rectangular Cartesian System of Coordinates or a method of associating points with numbers, and by doing so, associated a curve with its equation. Great progress in mathematics and the application of mathematics in science followed after this unification of algebra and geometry (Smoller). By definition of the Cartesian product of two sets, the case of interest is X ? Y where X and Y are both the set of real numbers R is symbolically denoted R x R ? { (x, y) | x ? R and y ? R }. Each member of the set is an ordered pair (x, y) and through the Cartesian coordinate system, it is possible to set up an association between this set of all ordered pairs (x, y) of R x R and the set of all points in the plane. Hence, the two-dimensional coordinate system becomes important in relating a point in a plane and a pair of real numbers which may be constructed using two perpendicular straight lines, vertical and horizontal, commonly known as the coordinate axes. With the point of intersection being the origin O, one may establish on each line a one-dimensional system which bears the same unit of length for both axes where, normally, the horizontal line refers to the x-axis or axis of the values of ‘x’ or abscissa whereas the vertical line pertains to the y-axis along which lie the values of ‘y’ called the ordinates. Once the axes are drawn, one can begin to plot a data of points (x, y) and in determining a point corresponding to an ordered pair of values, it helps to draw lines parallel to the axes through the point (x1, 0) on the x-axis and the point (0, y1) on the y-axis (Vance, 76). These lines intersect at a point P, a distance x1 from the y-axis (to the right or left, depending upon whether x1 is positive or negative) and a distance y1 from the x-axis (above or below, depending upon whether y1 is positive or negative). These distances can be called directed distances and the point P, determined by the ordered pair of values x1 and y1, is denoted by the ordered pair, expressed as (x1, y1), where x1 and y1 are called coordinates of P. The two coordinate axes divide the plane into four parts, called the first, second, third, and fourth quadrants. It is useful to verify that the coordinates of points located in the different quadrants have the signs shown in the table. Quadrants Abscissa Ordinate I + + II _ + III _ _ IV + _ Since every other point may be plotted on the xy-plane, the line or curve connecting the locus of points can be sketched to generate a graph from which to evaluate the behavior or profile of a function. In the presence of the Cartesian coordinate system, thus, there emerges a graphical method of solving a system of equations where each equation may be illustrated as linear, quadratic, rational, polynomial, exponential, or logarithmic in nature. From such two-dimensional coordinate system may be derived the ‘distance formula’ which has several uses to various areas in mathematics like analytic geometry whose foundations were primarily sought based on the Cartesian system of coordinates. To obtain an expression for the distance ‘d’ between any two points P1(x1, y1) and P2(x2, y2), where the same unit lengths are used to measure off from either axis, the Pythagorean theorem can be applied once the right triangle is formed out of the two points when the imaginary directed distances are made to enclose a figure with the third segment joining the points and labeled at the vertices for complete guide. This yields then to the distance formula for which a theorem states that ‘the distance between any two points P1(x1, y1) and P2(x2, y2) is given by: d = P1P2 = [ (x1 - x2)2 + (y1 - y2)2 ]1/2. Moreover, the right triangle produced over the Cartesian plane together with the distance formula provides an insight to graphing a circle and setting up its equation according to the information presented in the graph. For instance, giving expression for the distance between the origin and the point (x, y), and equating this distance to 1 is equivalent to finding an equation of a circle that is centered at the origin (0, 0) and with radius of 1 so that x2 + y2 = 1. This may be counterchecked for validity by plugging into it a pair of coordinates from the unit circle illustration to see if the point does satisfy the equation (Leithold). Other Implications of Descartes’ Contribution Applying the concept of distance through the Cartesian coordinate system implies extended utilization for other problems which may involve geometric approach of proving in analytic geometry. Other than locating the distance between two points in the graph, the two-dimensional system of coordinates by Descartes can be used as well to prove, for example, that two sides of a triangle are equal by showing that the triangle with vertices given is an isosceles triangle. Alternately, the proof may be required to exhibit that given points are vertices of an isosceles or an equilateral triangle. With Cartesian plane, one can further show that certain points belong to a parallelogram as its vertices or prove whether or not some other point set or loci are collinear by including algebraic proof. Working with a Cartesian graph, it is possible to come up with an equation for each of the conic sections – circle, ellipse, and hyperbola given radius, foci, center, and other properties associated with each equation type. If a curve passes through a point, the coordinates of such point must satisfy the equation representing the curve. Trigonometric functions including those encountered in calculus are also graphed on the xy-plane besides the simple linear functions and on the basis of inspecting the graph, one would be able to tell at which regions the function has increasing or decreasing behavior and which intervals possess values that are either the maximum or the minimum of a function. When a quadratic function is graphed on the Cartesian coordinate system, for instance, vertex point serves as the highest point when the parabola opens downward and would be the lowest point when the parabola opens upward. At y = 0, the graph crosses the x-axis and these values are called the x-intercepts or zeros or solutions of the sample quadratic function. If the parabolic curve, whether concave up or concave down, does not hit the x-axis then there exists imaginary or complex values for the x-intercepts when the quadratic formula is used to find the nature of discriminant (Leithold). One Unification Aspect indicates that “Our study of the implications of the Cartesian coordinate system corresponds to internal, mental activities; our use of this knowledge for practical applications involve external, physical actions” (New World). This statement occurs mathematically irrelevant yet in the study of mathematics, especially of the tools for higher math course, integrated aspects of mental and physical endeavor are indeed at work in guiding one’s skills to plot and understand coordinates and the relevance of the system which Descartes invented in order to connect the principles of algebra to those of Euclidean geometry. References Vance, Elbridge P. (1962). Modern College Algebra. Massachusetts: Addison – Wesley Publishing Company, Inc. Leithold, Louis. The Calculus with Analytic Geometry. 5th ed. Longman Higher Education. Smoller, Laura. “Did You Know ... ?” Applications: Web-Based Precalculus. Retrieved from http://ualr.edu/lasmoller/descartes.html on March 12, 2012. “Cartesian Coordinate System Images.” Retrieved from http://images.yourdictionary.com/cartesian-coordinate-system on March 12, 2012. New World Encyclopedia (2008). “Talk: Cartesian Coordinate System.” Retrieved from http://www.newworldencyclopedia.org/entry/Talk:Cartesian_coordinate_system on March 14, 2012. Read More