1. Chapter 8
Integers

2. 8.1 Addition and Subtraction
Definition: The set of integers is the set
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.

3. Set Model
In a set model, two different colored chips can be used, one color for positive numbers and another color for negative numbers.
--3 +5

4. 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.

5. All three examples represent +3.
Using Chips
Each integer has infinitely many representations using chips. +3 +3 +3
All three examples represent +3.

6. Number Line Representation
The integers are equally spaced and arranged symmetrically about 0.
Due to this symmetry, we have the concept of “opposite.” -5 -4 -3 -2 -1 1 2 3 4 5

7. Opposite Set Model: Measurement Model: -3 Opposites Opposites +3 -2 -11 2
Opposites
Opposites +3

8. Addition of Integers Definition: Let a and b be any integers.
If a and b are positive, they are added as whole numbers.
If a and b are positive (thus –a and –b are negative), then where a+b is the whole number sum of a and b.

9. Addition of Integers Continued
Adding a positive and a negative a. If a and b are positive and then where a – b is the whole number difference of a and b.
b. If a and b are positive and then
where b – a is the whole number difference of a and b.

10. Addition using the Set Model
Example:
Example: -3 +4 -3 -4 +1 -7

11. Properties Closure Property for Integer Addition.
Commutative Property for Integer Addition
Associative Property for Integer Addition
Identity Property for Integer Addition
Additive Inverse Property for Integer Addition

12. Additive Cancellation for Integers
Theorem: Let a, b, and c be any integers.
If then
Proof: Let Then
Addition
Associativity
Additive Inverse
Additive Identity

13. Theorem:
Let a be any integer. Then

14. Subtraction Pattern: The first column remains 4.
The second column decreases by 1 each time.
The column after the = increases by 1 each time.

15. Subtraction
Take-Away:
Take Away 3
Take Away 4 M
Leaves 2
Leaves –1

16. Subtraction Adding the Opposite Let a and b be any integers. Then
Missing-Addend Approach
Let a, b, and c be any integers. Then
if and only if

17. 8.2 Multiplication, Division, and Order
Positive Times a Negative
The first column remains 3.
The second column decreases by 1 each time.
The column after the = decreases by by 3 each time.

18. Negative Times a Positive
The first column remains –3 .
The second column decreases by 1 each time.
The column after the = increases by by 3 each time.

19. Chip Model
Positive Times a Negative
Combine 4 groups of 3 red chips

20. Negative Times a Positive
Chip Model
Negative Times a Positive
Take away 4 groups of 3 black chips.
Take away 4 groups of 3 blacks.
Insert 12 chips of each color.
Leaves 12 reds.
--12

21. Multiplication of Integers
Definition:
Let a and b be any integers.
If a and b are positive, they are multiplied as whole numbers.
If a and b are positive (thus–b is negative), then where is the whole number product of a and b.

22. Multiplication of Integers Continued
Multiplying two negatives a. If a and b are positive then where is the whole number product of a and b.

23. Properties Closure Property for Integer Multiplication
Commutative Property for Integer Multiplication
Associative Property for Integer Multiplication
Identity Property for Integer Multiplication
Distributive Property of Multiplication over Addition

24. Some Theorems Theorem: Let a be any integer. Then
Theorem: Let a and b be any integers. Then

25. Two More Properties
Multiplicative Cancellation Property Let a, b, c be any integers with If then
Zero Divisors Property Let a and b be integers. Then if and only if or a and b both equal zero.

26. Division
Definition: Let a and b be any integers, where Then if and only if
for a unique integer c.

27. Negative Exponents Definition: Negative Integer Exponent
Let a be any nonzero number and n be a positive integer.
Then

28. Scientific Notation
A number is said to be in scientific notation when expressed in the form where and n is any integer.
The number a is called the mantissa and the exponent n is the characteristic.

29. Ordering Integers Less Than: Number Line Approach
The integer a is less than the integer b, written if a is to the left of b on the integer number line. -5 -4 -3 -2 -1 1 2 3 4 5
Since –2 is to the left of 4 on the number line, --2 is less than 4.

30. Ordering Integers Less Than: Addition Approach Since –2 +6=4,
The integer a is less than the integer b, written if and only if there is a positive integer p such that
Since –2 +6=4,

31. Properties of Ordering Integers
Let a, b and c be any integers, p a positive integer and n a negative integer.
Transitive Property for Less Than
Property of Less than and Addition
Property of Less Than and Multiplication by a Positive.
Property of Less Than and Multiplication