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Math 40
Prealgebra
Section 2.1 – Introduction to Integers
2.1 Introduction to Integers
Recall
The Set of Natural Numbers
1, 2, 3, 4, 5, 6, ...
The Set of Whole Numbers
0, 1, 2, 3, 4, 5, 6, ...
The set of whole numbers includes zero while the set of natural numbers does not.
One way to help you remember the difference between these two sets is to think of the letter o in the word
“whole” as a zero.
The Set of Integers
..., 6,  5,  4,  3,  2,  1, 0, 1, 2, 3, 4, 5, 6, ...
The set of integers is the set of whole numbers combined with their opposites.
There are no fractions or decimal numbers in the set of integers.
Number Line
Zero is also known as the origin. Zero is neither positive nor negative.
Integers greater than zero are called positive integers.
Integers less than zero are called negative integers.
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Math 40
Prealgebra
Section 2.1 – Introduction to Integers
Notation
Positive integers can be written in two ways.
1) Write a positive sign in front of the integer:  5 “positive 5”
2) Do not write any sign: 5 is assumed to be “positive five”
Negative integers are written with a negative sign in front of the integer and can be
written in two ways.
1) Write a raised negative sign: —4 “negative 4” or “the opposite of 4”
2) Write the negative sign in the middle : 4
Graphing Integers
Integers are sometimes represented as points on a number line.
Example 1: Graph 3 and 3 on the number line below.
Solution: To graph 3 on a number line, we draw a closed circle at its location on the number line.
To graph 3 on a number line, we draw a closed circle at its location on the number line.
You Try It 1: Graph 7,  5, 4, 0, and  8 on the number line below.
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Math 40
Prealgebra
Section 2.1 – Introduction to Integers
Comparing Integers
Notation
= means “is equal to”
> means “greater than”
< means “less than”
Consider the graph of 5 and 8 :
5 is to the left of 8 so we say “ 5 is less than 8 ” and write 5  8 .
8 is to the right of 5 so we say “ 8 is greater than 5 ” and write 8  5 .
Now consider the graph of 7 and 2 :
7 is to the left of 2 so we say “ 7 is less than 2 ” and write 7  2 .
2 is to the right of 7 so we say “ 2 is greater than 7 ” and write 2  7 .
Example 2: Write < or > between each pair of integers to make a true statement.
a) 0
b) 7
9
6
c) 3
Solution: a) 0 is to the left of 9 on the number line, so use a less than symbol: 0
8

b) 7 is to the left of 6 on the number line, so use a less than symbol: 7
9

c) 3 is to the right of 8 on the number line, so use a greater than symbol: 3
6

8
You Try It 2: Write < or > between each pair of integers to make a true statement.
a) 6
9
b) 0
2
c) 1
4
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2015 Carreon
Math 40
Prealgebra
Section 2.1 – Introduction to Integers
Absolute Value
Absolute Value of an Integer
The absolute value of an integer is its distance from 0 on the number line.
Absolute value is indicated by enclosing the integer (or expression) using two vertical
bars
“the absolute value of 4 ” is written as 4
Example 3: Simplify each expression.
a)
4
b)
7
c)
0
Solution: Each integer is enclosed in vertical bars. This means we find the absolute value of each integer.
a) The integer 4 is 4 units from 0 on the number line so 4  4 .
4 units
b) The integer 7 is 7 units from 0 on the number line so  7  7 .
7 units
c) The integer 0 is 0 units from 0 on the number line so 0  0 .
0 units
Note: Distance is never a negative quantity. This means that the absolute value of an
integer can only be positive or zero.
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2015 Carreon
Math 40
Prealgebra
Section 2.1 – Introduction to Integers
You Try It 3: Simplify each expression.
a)
1
b)
16
c)
 20
Translating from English to Math
Let’s go back and look at the graph of Example 1:
Notice that 3 and 3 are on opposite sides of the origin. We say that 3 and 3 are opposites.
Translating from English to Math
There are two ways to verbally indicate that a negative sign is in front of the integer (or
expression):
1) “the opposite of”
2) “negative”
Example 4: Translate the following English phrases into a numerical expression and simplify if possible.
a) the opposite of 5
b) the opposite of 4
c) negative seventeen
d) the opposite of negative twelve
Solution:
English
a) the opposite of 5
b) the opposite of 4
c) negative seventeen
Expression
5
  4 
Math grammar: Use parentheses to
avoid two symbols in a row.
17
  12 
d) the opposite of negative twelve
Math grammar: Use parentheses to
avoid two symbols in a row.
Simplify
5
(already simplified)
4
The opposite of negative
4 is positive 4.
17
(already simplified)
12
The opposite of negative
12 is positive 12.
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Math 40
Prealgebra
Section 2.1 – Introduction to Integers
You Try It 4: Translate the following English phrases into a numerical expression and simplify if possible.
Complete the table below.
English
Expression
Simplify
a) the opposite of 15
b) the opposite of 20
c) the opposite of negative one
Example 5: Translate the following English phrases into a numerical expression and simplify if possible.
a) the absolute value of 9
b) the opposite of the absolute value of 9
c) the opposite of the absolute value of 9
d) the opposite of 9
Solution:
English
a) the absolute value of 9
Expression
Simplify
9
9
9
b) the opposite of the absolute
value of nine
 9
Find the absolute value of
9 first, then take the
opposite.
c) the opposite of the absolute
value of 9
 9
Find the absolute value of
negative 9 first, then take
the opposite.
9
  9 
d) the opposite of negative nine
Math grammar: Use parentheses to
avoid two symbols in a row.
9
The opposite of negative
12 is positive 12.
You Try It 5: Translate the following English phrases into a numerical expression and simplify if possible.
Complete the table below.
English
a) the opposite of the absolute
value of fourteen
Expression
Simplify
b) the opposite of 14
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