Chapter 1 Lesson 1_Ordered Pairs of Real Numbers.pptx
- 1. ORDERED PAIRS OF REAL NUMBERS Prepared by: Estelita A. Villa
- 2. ANALYTIC GEOMETRY Analytic Geometry deals with the properties, behaviors and solutions to points, lines, curves, angles, surfaces and solids by means of algebraic methods in relation to a coordinate system. Hence, this branch of Mathematics will tackle all about geometric figures as plotted on the rectangular coordinate system, otherwise known as the Cartesian coordinate system. Their solutions and proofs shall likewise be resolved using the concepts of Algebra.
- 3. THE REAL NUMBER LINE A basic concept of analytic geometry is the representation of all real numbers by points on a directed line. The real numbers consist of the positive numbers, negative numbers, and zero.
- 4. THE REAL NUMBER LINE Another basic concept is that of a line and a plane. Two points determine a line and three non-collinear points determine a plane.
- 5. To establish the desired representation, select a point O of the line, and call it the origin, to represent the number zero. Then, mark points at distances 1, 2, 3, and so on, units to the right of the origin. These points represent the numbers 1, 2, 3, and so on. In the same way, locate the points to the left of the origin to represent the numbers -1, -2, -3, and so on. Points are now assigned to the positive integers, the negative integers and the integer zero. Numbers whose values are between two consecutive integers have their corresponding points between the points associated with those integers. Thus, the number 2¼ corresponds to the point 2¼ units to the right of the origin. And, in general, any positive number p is represented by the point p units to the right of the origin, and a negative number q is represented by the point q units to the left of the origin. Assume that every real number corresponds to one point on the line and, conversely, every point on the line corresponds to one real number. This relation of the set of real numbers and the set of points on a directed line is called one-to-one correspondence.
- 6. The directed line with its points corresponding to real numbers, is called a real number line. The number corresponding to a point on the line is called the coordinate of the point. Since the positive numbers correspond to points in the chosen positive direction from the origin and the negative numbers correspond to points in the opposite or negative direction from the origin, then the coordinate of point on a number line is said to be the directed distance from the origin.
- 7. DIRECTED LINE SEGMENTS A line segment is measured in a definite sense from one endpoint to the other. Its direction is indicated by an arrowhead. The distance between two distinct points on a directed line segment is called directed distance. On the other hand, the undirected distance (∞, read as infinity) is the length of the segment in which no definite distance has been defined and that it is taken as positive. In addition, if the segment is read from left to right, then the segment has a positive magnitude; if right to left, it has a negative magnitude. Thus, the positive or negative sign indicates the direction from which the segment is taken.
- 8. DIRECTED DISTANCE (a) positive directed distance (b) negative directed distance
- 9. UNDIRECTED DISTANCE (c) positive undirected distance (d) negative undirected distance
- 10. FUNDAMENTAL PROPERTY OF DIRECTED LINE SEGMENT If A, B and C are three distinct points on a line (collinear), the following relation holds as a fundamental property of a directed line segment.
- 11. EXAMPLE If points A, B and C are collinear, find the values of the following:
- 12. CARTESIAN COORDINATE SYSTEM Cartesian coordinate system (also known as rectangular coordinate system) consists of two perpendicular lines which intersect at the origin. x-axis and y-axis are called coordinate axe coordinate plane
- 13. DEFINITIONS Definition 1.1: A pair of numbers (x, y) in which the order of occurrence of the numbers is distinguished is an ordered pair of numbers. Two ordered pairs, (x, y) and (x’, y’), are equal if and only if x = x’ and y = y’. Definition 1.2: The x-coordinate, or abscissa, of a point P is the directed distance from the y-axis to the point. The y-coordinate, or ordinate, of a point P is the directed distance from the x-axis to the point. The above definitions tell us that in an ordered pair, say (x,y), the first number (x) is called the abscissa and the second number (y) is called the ordinate of the point. Thus, given an ordered pair (2, 7), 2 is the abscissa
- 14. POSITION OF A POINT ON A SURFACE If a point lies on a given surface, two magnitude, or coordinates are necessary to determine its position, each coordinate being measured in a definite sense. To locate a point defined by an ordered pair, say (x,y) on the Cartesian coordinate system, the abscissa determines the number of units the point departs from the origin horizontally. If the abscissa is positive, then the point departs to the right; if it is negative, then it departs to the left. Consequently, the ordinate of the point determine the number of units the point departs from the origin vertically. If the ordinate is positive, then the point departs upward; if negative, it departs downward.
- 15. EXAMPLE Using the rectangular coordinate system, plot the following points: a. (1, 4) b. (-3, 2) c. (-5, -3) d. (5, -4) Solution: The graph is shown in the right figure. . . . . x y -x -y a (1, 4) d (5, -4) b (-3, 2) c (-5, -3)
- 16. TRY THESE A. Draw the figures defined by the following points on separate coordinate paper.