5-1
1 Integers and the Operations of Addition and Subtraction
Representations of Integers
Integer Addition
Number-Line Model
Absolute Value
Properties of Integer Addition
Integer Subtraction
Representations of Integers
The set of integers is denoted by I:
The negative integers are opposites of the positive integers.
–
4 is the opposite of positive 4
3 is the opposite of –3
Example 1 For each of the following, find the opposite of x.
a. x = 3
b. x = −5
c. x = 0
Integer Addition
Chip Model
Black chips represent positive integers and red chips represent negative
integers. Each pair of black/red chips neutralize each other.
Charged-Field Model
Similar to the chip model. Positive integers are represented by +’s and
negative integers by –’s.
’s. Positive charges neutralize negative charges.
Number-Line Model
Positive integers are represented by moving forward (right) on the number line; negative
integers are represented by moving backward (left).
Example 2 The temperature was −4°C. In an hour, it rose 10°C. What is the new
temperature?
Integer Addition
Pattern Model
Beginning with whole number facts, a table of computations is created by following a pattern.
Absolute Value
The absolute value of a number a, written |a|,, is the distance on the number line from 0 to a.
Definition
Example 3 Evaluate each of the following.
a. |20|
b. |−5|
c. |0|
d.
− −
| 3|
e. |2 + −5|
Properties of Integer Addition
Integer addition has all the properties of wholewhole number addition.
Properties of the Additive Inverse
By definition, the additive inverse, −a, is the solution of the equation x + a = 0.
For any integers a and b, the equation x + a = b has a unique solution, b + −a.
Example 4 Find the additive inverse of each of the following.
a. −(3 + x)
b. a + −4
c.
−
3 + −x
Integer Subtraction
Chip Model for Subtraction
To find 3 − −2, add 0 in the form 2 + −2 (two black chips and two red chips) to the three black
chips, then take away the two red chips.
Charged-Field Model for Subtraction
To find −3 − −5, represent −3 so that at least five negative charges are present. Then take away
the five negative charges.
Number-Line Model
While integer addition is modeled by maintaining the same direction and moving forward or
backward depending on whether a positive or negative integer is added, subtraction is modeled
by turning around.
Pattern Model for Subtraction
Using inductive reasoning and starting with known subtraction facts, find the difference of two
integers by following a pattern.
Subtraction Using the Missing Addend Approach
Subtraction of integers, like subtraction of whole numbers, can be defined in terms of addition.
Subtraction
For integers a and b, a − b is the unique integer n such that a = b + n.
Example 5 Use the definition of subtraction to compute the following:
a. 3 − 10
b.
−
2 − 10
Subtraction Using Adding the Opposite Approach
Subtracting an integer
eger is the same as adding its opposite.
Example 6 Using the fact that a − b = a + −b, compute each of the following:
a. 2 – 8
b. 2 – -8
c. -12 – -5
d. -12 – 5
Example 7 Write expressions equal to each of the following without parentheses.
a. –(b – c)
b. a – (b + c)
Example 8 Simplify each of the following.
a. 2 − (5 − x)
b. 5 − (x − 3)
c.
−
(x − y) – y
Order of Operations
Recall that subtraction is neither commutative nor associative.
An expression such as 3 − 15 − 8 is ambiguous unless we know in which order to perform the
subtractions.
Mathematicians agree that 3 − 15 − 8 means (3 − 15) − 8.
Subtractions are performed in order from left to right.
Example 9 Compute each of the following.
a. 2 − 5 − 5
b. 3 − 7 + 3
c. 3 − (7 − 3)
Homework 5.1 # A-1, 3, 5, 6, 8, 9, 12, 13, 15, 18, 20, 22, 24, 25, 27, 28, B-3, 6, 13, 17, 23, 25, 27