Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 1.1 Describing Data with Sets of...

Copyright © 2013, 2009, 2005 Pearson Education, Inc.

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 NumbersNatural 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

Solutiona. natural number, whole number, integer, rational number

b. integer, rational number

c. whole number, integer, rational number

8

4


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: 2, 15,


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.Solutiona. natural number, integer, rational number

b. rational number

c. irrational number

7


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?Solutiona. To find the average, we find the sum of the four test scores and divide by 4:

b. rational and real numbers

91 96 89 84 36090

4 4


, 0 and 1,xf x a a a

For any real number a,

a + 0 = 0 + a = a

and

a ·1 = 1 · a = a.

IDENTITY PROPERTIES

Properties of Real Numbers--Summary



For any real numbers a and b,

a + b = b + a

and

a ·b = b · a.

COMMUTATIVE PROPERTIES



For any real numbers a, b, and c,

(a + b) + c = a + (b + c)

and

(a ·b) · c = a · (b · c).

ASSOCIATIVE PROPERTIES


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

a(b c) = ab ac.

DISTRIBUTIVE PROPERTIES


Example

Apply a distributive property to each expression. a. 5(2 + y) b. 8 – (2 + w)c. 5x – 2x d. 3y + 4y – y

Solutiona. b.

c. d.

5( ) 52 2 5y y 10 5y

8 (2 ) 8 1 2 1w w 8 2 w 6 w

5 2 (5 2)x x x

3x

3 4 1 (3 4 1)y y y y 6y

Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 1.1 Describing Data with Sets of...

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