Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 1.1 Describing Data with Sets of Numbers
Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 1.1 Describing Data with Sets of Numbers.
Mar 26, 2015
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Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Section 1.1
Describing Data with Sets of
Numbers
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
Objectives
• Natural and Whole Numbers
• Integers and Rational Numbers
• Real Numbers
• Properties of Real Numbers
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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.
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. 8 c. 0
Solutiona. natural number, whole number, integer, rational number
b. integer, rational number
c. whole number, integer, rational number
8
4
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: 2, 15,
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. 8 b. 1.6 c.Solutiona. natural number, integer, rational number
b. rational number
c. irrational number
7
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
, 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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
, 0 and 1,xf x a a a
For any real numbers a and b,
a + b = b + a
and
a ·b = b · a.
COMMUTATIVE PROPERTIES
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
, 0 and 1,xf x a a a
For any real numbers a, b, and c,
(a + b) + c = a + (b + c)
and
(a ·b) · c = a · (b · c).
ASSOCIATIVE PROPERTIES
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
Copyright © 2013, 2009, 2005 Pearson Education, Inc.
, 0 and 1,xf x a a a
For any real numbers a, b, and c,
a(b + c) = ab + ac
and
a(b c) = ab ac.
DISTRIBUTIVE PROPERTIES
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
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
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