Name Period Date: Topic: Real Numbers and Their Graphs Standard: 9-12.A.1.3 Objective: Essential Question: What is the significance of a point on a number line? Determine the relative position on the number line and the relative magnitude of integers, decimals, rationals, irrationals, and numbers in scientific notation. The student will learn to graph real numbers on a number line, to compare numbers, and to find their absolute values. Natural Numbers: {1, 2, 3, } Also called the counting numbers The set of natural numbers is represented by the symbol,. Whole Numbers: {0, 1, 2, 3, } The set of whole numbers is the union of the set of natural numbers and the set containing only zero: * +. Integers: {,-3, -2, -1, 0, 1, 2, 3, } The set of integers is represented by the symbol. Summary
Rational Numbers: { } e.g. 1,, 0.5,, 0 The set of rational numbers is represented by the symbol, (for quotient). Irrational Numbers:.{ } e.g.,, Real Numbers: The set of real numbers is the union of the set of rational numbers and the set of irrational numbers. The set of real numbers is represented by the symbol,. Value of Real Numbers: The value, or magnitude, of a real number can be represented by placing a dot on a number line. Number Line: A number line is a horizontal line divided into equal segments by short, vertical lines. Each vertical line represents a real value. We usually represent integral values by the vertical lines. We usually indicate the position of zero by a larger line, and we call that location the origin. 2
Graphing Real Numbers: Each point on a number line is paired with exactly one real number. That number is called the coordinate of the point. The dots on the number line below have coordinates, 3.1,, 1, and. Each real number is paired with exactly one point on the number line. That point is called the graph of the point. The dots on the number line below are the graphs of the real numbers, 3.1,, 1, and. origin -3.1 1 Exercise: Graph the following points on the number line below:,,,. 3
Exercise: What are the coordinates of points A, B, and C on the number line below? C B A A: B: C: 4
Distance between points on a number line: The distance between two points on a number line is the algebraic difference between their coordinates. What is the coordinate of a point that is of the way from point C, below, to point D? C P D Coordinate of point D: 2 Coordinate of point C: 3 Distance between D and C: ( ) Two fifths of distance Coordinate of point is, ( ) This point is indicated by point P above. 5
Exercise: What is the coordinate of a point that is of the way from point C, below, to point D? C D Graph the point. What is the coordinate of a point that is of the way from point C, below, to point D? C D Graph the point. 6
Order on a Number Line: On a number line, the larger a number is, the farther to the right its graph is. C D The number line clearly shows that negative three is smaller than two. Moreover, two is larger than negative three. The order of two numbers, as seen on a number line, can be expressed as an inequality. Inequalities: In the above example, we can write, This is read, Negative three is less than two. Moreover, the relationship between these two numbers can also be written as, These two inequalities are equivalent... 7
Exercise: Use the number line to fill in the blank in the expressions below with the correct inequality symbol. 2 1 2 1 3 0 1 2 1 3 0 1 0 3 2 3 8
Opposite of a Real Number: Every real number has an opposite. Opposite real numbers lie equidistant from and on opposite sides of the origin. 2 2 The opposite of two is negative two. ( ) The opposite of negative two is two. ( ) Exercise: On the number line below, graph and label the following points: The opposite of one ( ) The opposite of two ( ) The opposite of negative three ( ) The opposite of zero ( ) Zero is its own opposite: ( ) 9
Absolute Value of a Real Number: The absolute value of a real number is the distance between the graph of that number and the origin. 2 3 The absolute value of three is three, because the distance between the origin and three is three. The absolute value of negative two is two, because the distance between the origin and negative two is two. Symbolically, absolute value is written using two vertical lines: Definition of Absolute Value: x x x x x x x x x 10
Exercise: On the number line below, graph and label the following points: Class work: Page 3 Oral Exercises 1-19 Homework: Page 4 Written Exercises 1-39 odd Page 5 Mixed Review 1-12 11