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Chapter 1
Introduction to
Algebraic
Expressions
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1.4
Positive and Negative
Real Numbers
• Whole Numbers and Integers
• The Rational Numbers
• Real Numbers and Order
• Absolute Value
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Whole Numbers and Integers
• A set is a collection of objects.
• The integers consist of all whole numbers and
their opposites.
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Set of Integers
The set of integers =
{…−4, −3, −2, −1, 0, 1, 2, 3, 4, …}
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Example
State which integer(s) corresponds to the situation.
• The lowest point in Australia is Lake Eyre at 15 m
below sea level and the highest point is Mt.
Kosciuszko at 2229 m above sea level.
Solution
The integer −15 corresponds to 15 m below sea
level.
The integer 2229 corresponds to 2229 m above sea
level.
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1-5
Set of Rational Numbers
The set of rational numbers =
a

 | a and b are integers and b  0  .
b

This is read “the set of all numbers a over b,
where a and b are integers and b does not
equal zero.”
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Example
3
Convert to decimal notation: .
8
Solution
0.375
8 3.000
24
60
56
40
40
0
← The remainder is 0.
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Example
5
.
Convert to decimal notation:
22
0.2272
Solution
22 5.000
44
60
44
160
154
5
 0.227
22
60
44
16
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1-8
Set of Real Numbers
The set of real numbers = The set of all
numbers corresponding to points on the
number line.
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Example
Which numbers in the following list are (a) whole numbers?
(b) integers? (c) rational numbers? (d) irrational numbers?
(e) real numbers?
9
27,
, 0, 4, 65, 40, 72
5
Solution
a) whole numbers: 0, 4, 65, 72
b) integers: −27, 0, 4, 65, 72
9

27,
, 0, 4, 65, 72
c) rational numbers:
5
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Example
Which numbers in the following list are (a) whole numbers?
(b) integers? (c) rational numbers? (d) irrational numbers?
(e) real numbers?
9
27,
, 0, 4, 65, 40, 72
5
Solution
d) irrational numbers:
e) real numbers: 27,
40
9
, 0, 4, 65,
5
40, 72
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Real Numbers
• Real numbers are named in order on the
number line, with larger numbers further to the
right.
• < mean “less than”
• > means “greater than”
−5 < 8
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
−2 > −9
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Example
Use either < or > for the blank to write a true
statement. (a) −3.42 ____ 2.35
(b) 7 ____ −15
(c) −11 ____ −9
Solution
a) Since −3.42 is to the left of 2.35, we have
−3.42 < 2.35.
b) Since 7 is to the right of −15, we have 7 > −15.
c) Since −11 is to the left of −9, we have −11 < −9.
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More Inequalities
≤
means “is less than or equal to”

means “is greater than or equal to”
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Example
Classify each inequality as true or false.
a) −9 ≤ 7
b) −8  −8
c) −7  −2
Solution
a) −9 ≤ 7 is true because −9 < 7.
b) −8  −8 is true because −8 = −8.
c) −7  −2 is false since −7 > −2 nor −7 = −2 is
true.
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Absolute Value
We write |a|, read “the absolute value of a,”
to represent the number of units that a is
from zero.
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Example
Find the absolute value:
a. |–4|
b. |3.8|
c. |0|
Solution
a) |–4| = 4 since –4 is 4 units from 0.
b) |3.8| = 3.8 since 3.8 is 3.8 units from 0.
c) |0| = 0 since 0 is 0 units from itself.
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