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**Describing Data with Sets of Numbers**

Section 1.1 Describing Data with Sets of Numbers

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**Objectives Natural and Whole Numbers Integers and Rational Numbers**

Real Numbers Properties of Real Numbers

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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.

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Example Classify each number as one or more of the following: natural number, whole number, integer, or rational number. a. b. 8 c. 0 Solution a. natural number, whole number, integer, rational number b. integer, rational number c. whole number, integer, rational number

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**Real Numbers: Can be represented by decimal numbers**

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:

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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. 8 b. 1.6 c. Solution a. natural number, integer, rational number b. rational number c. irrational number

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Example A student obtains the following test scores: 91, 96, 89, and 84. a. Find the student’s 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

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**Properties of Real Numbers--Summary**

IDENTITY PROPERTIES For any real number a, a + 0 = 0 + a = a and a ·1 = 1 · a = a.

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**For any real numbers a and b, a + b = b + a and a ·b = b · a.**

COMMUTATIVE PROPERTIES For any real numbers a and b, a + b = b + a and a ·b = b · a.

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**For any real numbers a, b, and c, (a + b) + c = a + (b + c) and **

ASSOCIATIVE PROPERTIES For any real numbers a, b, and c, (a + b) + c = a + (b + c) and (a ·b) · c = a · (b · c).

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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

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**For any real numbers a, b, and c, a(b + c) = ab + ac and **

DISTRIBUTIVE PROPERTIES For any real numbers a, b, and c, a(b + c) = ab + ac and a(b c) = ab ac.

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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.

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