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Copyright 2013, 2009, 2005 Pearson Education, Inc. Section 1.1 Describing Data with Sets of Numbers Slide 2 Copyright 2013, 2009, 2005 Pearson Education, Inc. Objectives Natural and Whole Numbers Integers and Rational Numbers Real Numbers Properties of Real Numbers Slide 3 Copyright 2013, 2009, 2005 Pearson Education, Inc. Types of Numbers Natural Numbers: The set of counting numbers. N = {1, 2, 3, 4, 5, 6, } Set braces, { }, are used to enclose the elements of a set. Whole Numbers: W = {0, 1, 2, 3, 4, 5, } Integers: I = {, 3, 2, 1, 0, 1, 2, 3, } Rational Number: any number that can be expressed as the ratio of two integers; p/q, where q is not equal to 0 because we cannot divide by 0. Slide 4 Copyright 2013, 2009, 2005 Pearson Education, Inc. Example Classify each number as one or more of the following: natural number, whole number, integer, or rational number. a. b. 8c. 0 Solution a. natural number, whole number, integer, rational number b. integer, rational number c. whole number, integer, rational number Slide 5 Copyright 2013, 2009, 2005 Pearson Education, Inc. Real Numbers: Can be represented by decimal numbers. Every fraction has a decimal form, so real numbers include rational numbers. Irrational Numbers: A number that cannot be expressed by a fraction, or a decimal number that does not repeat or terminate. Examples: Slide 6 Copyright 2013, 2009, 2005 Pearson Education, Inc. Example Classify each real number as one or more of the following: a natural number, an integer, a rational number, or an irrational number. a. 8b. 1.6c. Solution a. natural number, integer, rational number b. rational number c. irrational number Slide 7 Copyright 2013, 2009, 2005 Pearson Education, Inc. Example A student obtains the following test scores: 91, 96, 89, and 84. a. Find the students average test score. b. Is this average a natural, rational, or a real number? Solution a. To find the average, we find the sum of the four test scores and divide by 4: b. rational and real numbers Slide 8 Copyright 2013, 2009, 2005 Pearson Education, Inc. For any real number a, a + 0 = 0 + a = a and a 1 = 1 a = a. IDENTITY PROPERTIES Properties of Real Numbers--Summary Slide 9 Copyright 2013, 2009, 2005 Pearson Education, Inc. For any real numbers a and b, a + b = b + a and a b = b a. COMMUTATIVE PROPERTIES Slide 10 Copyright 2013, 2009, 2005 Pearson Education, Inc. For any real numbers a, b, and c, (a + b) + c = a + (b + c) and (a b) c = a (b c). ASSOCIATIVE PROPERTIES Slide 11 Copyright 2013, 2009, 2005 Pearson Education, Inc. Example State the property of real numbers that justifies each statement. a. 5 (2x) = (5 2)x b. (1 3) 6 = 3 6 c. 7 + xy = xy + 7 Associative property for multiplication Identity property of 1. Commutative property for addition Slide 12 Copyright 2013, 2009, 2005 Pearson Education, Inc. For any real numbers a, b, and c, a(b + c) = ab + ac and a(b c) = ab ac. DISTRIBUTIVE PROPERTIES Slide 13 Copyright 2013, 2009, 2005 Pearson Education, Inc. Example Apply a distributive property to each expression. a. 5(2 + y)b. 8 (2 + w) c. 5x 2x d. 3y + 4y y Solution a.b. c.d.