# Describing Data with Sets of Numbers

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Describing Data with Sets of Numbers
Section 1.1 Describing Data with Sets of Numbers

Objectives Natural and Whole Numbers Integers and Rational Numbers
Real Numbers Properties of Real Numbers

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.

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

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:

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

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

Properties of Real Numbers--Summary
IDENTITY PROPERTIES For any real number a, a + 0 = 0 + a = a and a ·1 = 1 · a = a.

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.

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).

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

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.

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.