Chapter 1 Section 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Chapter Chapter 11Section Section 44

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


Real Numbers and the Number Line

11

44

33

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1.41.41.41.4Classify numbers and graph them on number lines.Tell which of two real numbers is less than the other.Find additive inverses and absolute values of real numbers.Interpret the meanings of real numbers from a table of data.


Objective 11

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Classify numbers and graph them on number lines.


Classify numbers and graph them on a number line.

The natural numbers and the whole numbers, along with many others, can be represented on a number line like the one below.

We draw a number line by choosing any point on the line and labeling it 0. Then we choose any point to the right of 0 and label it 1. The distance between 0 and 1 gives a unit of measure used to locate other points.

The “arrowhead” is used to indicate the positive direction on a number line.

Slide 1.4- 4


Classify numbers and graph them on a number line. (cont’d)

The natural numbers are located to the right of 0 on the number line. For each natural number, we can place a corresponding number to the left of 0. Each is the opposite, or negative, of a natural number.

Positive numbers and negative numbers are called signed numbers.

The natural numbers, their opposites, and 0 form a new set of numbers called the integers.

{. . . , −3, −2, −1, 0, 1, 2, 3, . . .}

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Use an integer to express the number in boldface italics in each application.

Erin discovers that she has spent $53 more than she has in her checking account.

The record-high Fahrenheit temperature in the United States was 134° in Death Valley, California, on July 10, 1913. (Source: World Almanac and Book of Facts 2006.)

EXAMPLE 1Using Negative Numbers in Applications

Solution: −53

Solution: 134

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{ is a quotient of two integers, with denominator not 0} is the set of rational numbers.

(Read as “the set of all numbers x such that x is a quotient of two integers, with denominator not 0.”)

A decimal number that comes to an end (terminates), such as

0.23 is a rational number. For example, 0.23 .

Since any integer can be written as a quotient of itself and 1, all integers are also rational numbers.

Example:

x x

55

1

23

100

This is called set-builder notation. This notation is convenient to use when it is not possible to list all the elements of a set.

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To graph a number, we place a dot on the number line at the point that corresponds to the number. The number is called the coordinate of the point.

{ is a nonrational number represented by a point on the number} is the set of irrational numbers.

The decimal form of an irrational number neither terminates nor repeats.

x x

{ is a rational or an irrational number} is the set of real numbers.

x x

Decimals numbers that repeat in a fixed block of digits, such

as 0.3333… , are also rational numbers. Example: 0.3 10.3

3

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EXAMPLE 2

Identify each real number in the set

as rational or irrational.

Solution:

Determining whether a Number Belongs to a Set

5 3, 7,1 ,0, 11,

8 5

5 7 3 8, 7 , 1 ,

8 1 5 5or or

and are rational;0

01

or

and π are irrational11

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Objective 22

Tell which of two real numbers is less than the other.

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For any two real numbers a and b, a is less than b if a is to the left of b on the number line.

Ordering of Real Numbers

This means that any negative number is less than 0, and any negative number is less than any positive number. Also, 0 is less than any positive number.

We can also say that, for any two real numbers a and b, a is greater than b, if a is to the right of b on the number line.

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EXAMPLE 3

Solution: False

Determining the Order of Real Numbers

Determine whether the statement is true or false.

1

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Objective 33

Find additive inverses and absolute values of real numbers.

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By a property of the real numbers, for any real number x (except 0), there is exactly one number, called the additive inverse, on the number line the same distance from 0 as x, but on the opposite side of 0.

Finding additive inverses and absolute values of real numbers.

The additive inverse of a number can be indicated by writing the − symbol in front of the number.

The additive inverse of −7 is written −(−7) and can be read “the opposite of −7” or “the negative of −7”

The Double Negative Rule, states that for any real number x, −(−x) = x

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Distance is a physical measurement, which is never negative. Therefore, the absolute value of a number is never negative.

Finding additive inverses and absolute values of real numbers. (cont’d)

The absolute value of a real number can be defined as the distance between 0 and the number on the number line. The symbol for the absolute value of the number x is |x|, read “the absolute value of x.”

The rule of Absolute Value says that, for any real number x,

.

0

0

x if xx

x if x

The “−x ” in the second part of the definition does not represent a negative number. Since x is negative in the second part, −x represents the opposite of a negative number—that is, a positive number.

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EXAMPLE 4

Solution:

Finding the Absolute Value

Simplify by finding the absolute value.

32 2 30

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Objective 44

Interpret the meanings of real numbers from a table of data.

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EXAMPLE 5

Solution: Iron and steal, from 2003 to 2004

Interpreting Data

In the table, which commodity in which year represents the greatest percent increase?

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Chapter 1 Section 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley