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Chapter 8
Integers
8.1 Addition and Subtraction
Definition: The set of integers is the set
I  ,3,2,1,0,1,2,3,.
The numbers 1, 2, 3, … are called positive
integers and the numbers -1, -2, -3, …
are called negative integers. Zero is
neither a positive or a negative integer.
Set Model
In a set model, two different colored chips
can be used, one color for positive
numbers and another color for negative
numbers.
+5
--3
Using Chips
One black chip represents a credit of one and one
red chip represents a debit of 1. One chip of
each color cancel each other out making 0.
0
Using Chips
Each integer has infinitely many representations
using chips.
+3
+3
+3
All three examples represent +3.
Number Line Representation
The integers are equally spaced and
arranged symmetrically about 0.
-5
-4
-3
-2
-1
0
1
2
3
4
5
Due to this symmetry, we have the concept
of “opposite.”
Opposite
Set Model:
Measurement
Model:
-3
-2
-1
0
1
Opposites
Opposites
+3
2
Addition of Integers
Definition:
Let a and b be any integers.
1. a  0  0  a  a
2. If a and b are positive, they are added as whole
numbers.
3. If a and b are positive (thus –a and –b are
negative), then  a    b   a  b 
where
a+b is the whole number sum of a and b.
Addition of Integers Continued
4. Adding a positive and a negative
a. If a and b are positive and a  b then
a   b   a  b where a – b is the whole
number difference of a and b.
5. b. If a and b are positive and a  b then
a   b   b  a  where b – a is the whole
number difference of a and b.
Addition using the Set Model
Example:  3   4
-3
-4
Example:
 3  4
-3
+4
+1
-7
Properties
1. Closure Property for Integer Addition.
2. Commutative Property for Integer
Addition
3. Associative Property for Integer Addition
4. Identity Property for Integer Addition
5. Additive Inverse Property for Integer
Addition
Additive Cancellation for Integers
Theorem: Let a, b, and c be any integers.
If a  c  b  c, then a  b.
Proof: Let a  c  b  c. Then
a  c    c   b  c    c 
a  c   c   b  c   c 
a0  b0
ab
Addition
Associativity
Additive Inverse
Additive Identity
Theorem:
Let a be any integer. Then   a  a.
Subtraction
Pattern:
The first column
remains 4.
The second
column decreases
by 1 each time.
422
4 1 3
40  4
4   1  5
4   2  6
The column
after the =
increases by 1
each time.
Subtraction
Take-Away:
34
53
Take Away 3
Take Away 4
Leaves 2
53  2
Leaves –1
3  4  1
Subtraction
Adding the Opposite Let a and b be
any integers. Then a  b  a   b.
Missing-Addend Approach
Let a, b, and c be any integers. Then
a  b  c if and only if a  b  c.
8.2 Multiplication, Division, and
Order
Positive Times a Negative
The first column
remains 3.
The second
column decreases
by 1 each time.
3  4  12
33  9
32  6
3 1  3
30  0
3   1  3
3   2  6
The column
after the =
decreases by
by 3 each
time.
Negative Times a Positive
 3  4  12
 3  3  9
 3  2  6
The second
column decreases  3  1  3
by 1 each time.
 3  0  0
 3   1  3
 3   2  6
The first column
remains –3 .
The column
after the =
increases by
by 3 each
time.
Chip Model
Positive Times a Negative
4   3  12
Combine 4 groups of 3 red chips
4   3  12
Chip Model
Negative Times a Positive
 4  3  12
Take away 4 groups of 3 black chips.
Take away 4
groups of 3
blacks.
Insert 12 chips of each color.
0
0
Leaves 12 reds.
--12
Multiplication of Integers
Definition:
Let a and b be any integers.
1.
a0  a0  0
2. If a and b are positive, they are multiplied as
whole numbers.
3. If a and b are positive (thus–b is negative), then
a b   ab  where ab is the whole number
product of a and b.
Multiplication of Integers
Continued
4. Multiplying two negatives
a. If a and b are positive then
 a b  ab where ab is the whole
number product of a and b.
Properties
1. Closure Property for Integer
Multiplication
2. Commutative Property for Integer
Multiplication
3. Associative Property for Integer
Multiplication
4. Identity Property for Integer Multiplication
5. Distributive Property of Multiplication
over Addition
Some Theorems
Theorem: Let a be any integer. Then
a1  a
Theorem: Let a and b be any integers.
Then
1.  ab  ab
2.  a  b   ab
Two More Properties
6. Multiplicative Cancellation Property
Let a, b, c be any integers with c  0
If ac  bc, then a  b.
7. Zero Divisors Property
Let a and b be integers. Then ab  0
if and only if a  0 or b  0 or a and b both
equal zero.
Division
Definition: Let a and b be any integers,
where b  0. Then a  b  c if and only if
a  b  c for a unique integer c.
Negative Exponents
Definition:
Negative Integer
Exponent
Let a be any nonzero
number and n be a
positive integer.
Then
a  aaa
a2  a  a
1
a a
3
a 1
0
1
a 
a
1
2
a  2
a
1
3
a  3
a
1
1
a  n
a
n
a
a
a
a
a
a
Scientific Notation
A number is said to be in scientific notation
when expressed in the form a  10
where 1  a  10 and n is any integer.
The number a is called the mantissa and
the exponent n is the characteristic.
n
Ordering Integers
Less Than: Number Line Approach
The integer a is less than the integer b, written a  b,
if a is to the left of b on the integer number line.
-5
-4
-3
-2
-1
0
1
2
3
Since –2 is to the left of 4 on the number
line, --2 is less than 4.
2 4
4
5
Ordering Integers
Less Than: Addition Approach
The integer a is less than the integer b, written a  b,
if and only if there is a positive integer p such that a  p  b.
Since –2 +6=4,  2  4.
Properties of Ordering Integers
Let a, b and c be any integers, p a positive integer and n a
negative integer.
1. Transitive Property for Less Than
If a  b and b  c, then a  c.
2.
Property of Less than and Addition
If a  b, then a  c  b  c.
3.
Property of Less Than and Multiplication by a Positive.
If a  b, then ap  bp.
4.
Property of Less Than and Multiplication
If a  b, then an  bn.