Lesson 1-2
Lesson
Vocabulary
Positive and Negative
Numbers
1-2
integers
positive integers
negative integers
positive numbers
negative numbers
BIG IDEA
Situations with two opposite directions lead
naturally to uses of negative numbers and zero.
inequality symbols
< (is less than)
> (is greater than)
Situations with Negative Numbers
double inequality
With the exception of 0, all the numbers mentioned in Lesson 1-1 are
positive numbers. Many situations lead to negative numbers and zero
as well as positive numbers.
Mental Math
Evaluate.
a. 7 · 5
Activity
b. 7 · 50
Step 1 Below are 16 phrases. Which would be described with positive
numbers, which with negative numbers, and which with zero?
Rearrange the phrases into 3 lists.
c. 70 · 500
d. 7,000 · 50
profit of
$3.21 billion
500 feet above
sea level
loss of
$4.6 million
282 feet below
sea level
sea level
7 seconds ago
23ºF below zero
60 seconds
from now
behind
10 points
no transaction
broke even
withdrawal of
$694.55
90ºF
deposit of $29
ahead 3 points
now
Step 2 Here are six situations: business, temperature, elevation, bank,
football, and time. Each situation uses 2 or 3 of the 16 phrases.
Which phrases go with each situation?
There are three common ways in which the “–” sign for negatives
is spoken.
Written
Spoken
Usage
–3
“negative 3”
correct
–3
“the opposite of 3”
correct
“minus 3”
commonly used, but can be confusing
since there is no subtraction
–3
See Quiz Yourself 1 at the right.
QUIZ YOURSELF 1
Translate –6 into words in
two different correct ways.
Positive and Negative Numbers
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Chapter 1
To enter a negative number into a graphing calculator, you can
usually use the opposite key, _. For example, to enter the number
–8 into the graphing calculator, key in _ 8. On some calculators, you
can use the subtraction symbol -.
Graphing Negative Numbers on a Number Line
On a horizontal number line, negative numbers are almost always
placed at the left. The numbers identified on the number line below
are the integers. The positive integers are the numbers 1, 2, 3, … .
The negative integers are –1, –2, –3, … . Zero is an integer but it is
neither positive nor negative. On the number line below, all numbers
to the right of 0 are positive numbers. All numbers to the left of 0
are negative numbers.
Negative numbers
᎑10 ᎑9
᎑8
᎑7
᎑6
᎑5 ᎑4
Zero
᎑3
᎑2
᎑1
0
Positive numbers
1
2
3
4
5
6
7
8
9
10
See Quiz Yourself 2 at the right.
Comparing Positive and Negative Numbers
Numbers are frequently compared. You can compare numbers
whether they are positive, negative, or zero. You may recall
comparing positive numbers with the inequality symbols < and
>. The symbol < means is less than and the symbol > means is
greater than. These same symbols are used with negative numbers.
QUIZ YOURSELF 2
What is the difference
between the whole
numbers and the positive
integers?
Example 1
Much of the city of New Orleans is situated about 8 feet
below sea level. Parts of Amsterdam, the largest city in the
Netherlands, are about 18 feet below sea level. Compare these
numbers using an inequality symbol.
Amsterdam
Solution Both cities are below sea level. The more you
descend from sea level, the lower the elevation is. Amsterdam is
more feet below sea level than New Orleans; therefore, it is the
lower city.
–18 < –8
You could also write this inequality as –8 > –18.
12
sand dunes above 1 meter
0-1 meter
below sea level
above 1 meter
Reading and Writing Numbers
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Lesson 1-2
Vertical number lines are also used to graph numbers. On vertical
number lines, positive numbers are usually placed above 0 and
negative numbers are placed below 0.
Example 2
The table below shows the low temperatures on February 28, 2005, for
three U.S. cities. Graph these numbers on a vertical number line.
City
Temperature
Boston, MA
24°F
Marquette, MI
–4°F
Duluth, MN
30
Boston
20
–13°F
Solution On the vertical number line at the right, the positive numbers are
above 0 and the negative numbers are below 0. Notice that on this number
line, each tick mark represents a distance of 10 degrees. This is the scale of
the number line.
10
0
᎑10
Marquette
Duluth
᎑20
You can also graph these numbers on a horizontal number line.
Duluth Marquette
᎑20
᎑10
Boston
0
10
20
30
Double Inequalities
Numbers graphed on a number line are also easy to compare, since
lesser numbers are usually to the left of or below greater numbers.
When the numbers are in order, inequalities can be combined.
The three temperatures of Example 2 can be compared using
two inequalities.
–13 < –4 < 24
Because this has two inequality signs, it is called a double
inequality. It can be read: “negative thirteen is less than negative
four, which is less than twenty-four.” You could also write:
24 > –4 > −13.
This is read: “twenty-four is greater than negative 4, which is greater
than negative thirteen.”
Caution: Do not use > and < in the same double inequality. For
example, do not write 24 > –13 < –4. Although this statement seems
correct at first glance, it is not true. In order for it to be true, each
individual inequality must be true. Here are the four inequalities in
the statement. Whether they are true or false is written below them.
24 > –13
–13 < –4
24 > –4
24 < –4
True
True
True
False
Positive and Negative Numbers
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Chapter 1
Questions
COVERING THE IDEAS
1. Give three examples of negative numbers from the newspaper
activity in Lesson 1-1.
2. Write an integer that could be used to represent each situation.
a. You make a savings-account withdrawal of $40.
b. You make a savings-account deposit of $60.
c. You owe $50.
d. Your team is behind by 5 points.
e. The game is tied.
3. Fill in the Blanks
a. On a horizontal number line, negative numbers are usually to
?
of the positive numbers.
the
b. On a vertical number line, positive numbers are usually
?
the negative numbers.
4. What is the scale of a number line?
In 5–7, rewrite the sentence using the symbol < or >.
5. –10 is less than –8.5.
6. –5 is greater than –5.7.
7. An elevation of 38 feet below sea level is lower than an elevation
of 25 feet above sea level.
8. Write in words: 79 > 0 > –16.
9. Multiple Choice Which of the following numbers is not
an integer?
A 5
B 2.5
C 0
D –3
APPLYING THE MATHEMATICS
10. Give an example of a number that is neither positive
nor negative.
11. Suppose time is measured in hours and 0 stands for the time
right now.
a. What number stands for an hour ago?
b. What number stands for two hours from now?
12. Give an example of a number that is negative but not an integer.
13. Tell whether each number is or is not a positive integer.
1
b. 0
c. 8
d. __22
e. 1
a. __2
14
Reading and Writing Numbers
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Lesson 1-2
14. Here is a list of some countries in Africa with their lowest points,
measured in meters.
Location
Elevation
Chott Melrhir, Algeria
40 m below sea level
Mahoun River, Burkina Faso
200 m above sea level
Gulf of Guinea, Cote d’Ivoire
at sea level
Kulul, Eritrea
75 m below sea level
Shire River, Malawi
37 m above sea level
Sebkha, Morocco
55 m below sea level
Great Usutu River, Swaziland
21 m above sea level
a. Draw a number line with a scale of 20 meters, and place the
elevations on it.
b. Write a double inequality relating the lowest points of
Burkina Faso, Eritrea, and Morocco.
15. In golf, par is the expected number of strokes to play a hole. A
golfer’s score is often kept by the total number of strokes under
or over par, rather than by the total number of strokes. The
player with the fewest strokes wins. The vertical number line
at the right shows the scores of some players at the end of the
fourth round of the 2006 LPGA Championship.
a. Se Ri Pak and Karrie Webb tied for first. Mi Hyun Kim and
Ai Miyazato tied for third. Represent their scores above or
below par as integers.
b. Here are the scores for other golfers. Copy the vertical
number line and place their scores on it.
Golfer
Score
Karen Stupples
9 over par
Marisa Baena
1 over par
Lorena Ochoa
5 under par
Nancy Scranton
᎑8
᎑7
᎑6
᎑5
᎑4
᎑3
᎑2
᎑1
0
1
2
1st
3rd
Par
par
c. Pak used 280 strokes in four rounds. How many strokes did
Stupples use?
16. Manuel wrote the double inequality –15 < 10 > 2. Write a note
to Manuel explaining why his inequality is incorrect. Then write
a correct double inequality relating the numbers.
Positive and Negative Numbers
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Chapter 1
REVIEW
17. In 2005, the population of Indonesia was
estimated to be about 242 million. Write
this number in base 10. (Lesson 1-1)
18. The country of Indonesia is an archipelago
(ar kuh PEH luh go) of 17,508 islands
lying between the Indian Ocean and the
Pacific Ocean. Write this number of islands
in words. (Lesson 1-1)
19. Multiple Choice Which of the following is
equal to 45? (Previous Course)
A 4+4+4+4+4
B 4×5
C 4×4×4×4×4
D 5×5×5×5
20. The sum of two whole numbers is 25. (Previous Course)
a. What is the greatest their product can be?
b. What is the least their product can be?
EXPLORATION
Indonesia includes the chain
of islands that curves along
the bottom of the photo and
all of the islands in the lower
right. The large island near the
center is shared by Indonesia,
Malaysia, and Brunei.
21. Question 14 gives the lowest points in seven countries in Africa.
Find the highest points in these countries and graph them on a
number line.
22. You are familiar with the Celsius and Fahrenheit temperature
scales. There is a third scale called the Kelvin scale. Zero on the
Kelvin scale is called “absolute zero.” Why is this called absolute
zero? What is absolute zero on the Celsius scale? What is it on
the Fahrenheit scale?
QUIZ YOURSELF ANSWERS
1. negative six, the opposite
of six
2. 0 is a whole number but
not a positive integer.
16
Reading and Writing Numbers
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