1. OPERATIONS ON INTEGERS
MSJC ~ San Jacinto Campus
Math Center Workshop Series
Janice Levasseur

2. Basic Definitions
Natural Numbers are the counting numbers: {1, 2, 3, 4, 5, 6, . . .}
Whole Numbers are the set of natural numbers with zero included: {0, 1, 2, 3, 4, 5, . . .}
Integers are the set of all whole numbers and their opposites: { , -2, -1, 0, 1, 2, 3, . . .}

3. Addition of Integers Ex: Consider the addition 3 + 2
We can illustrate the addition using hollow dots for positive numbers = 3 + 2 5
We conceptually understand the gathering up of like items to find the total.

4. Ex: Consider the addition -3 + (-2)
Similarly, we can illustrate the addition using solid dots for negative numbers = -3 + -2 -5
We again conceptually understand the gathering up of like items to find the total.

5. But, what does 3 + (-2) mean? How can we illustrate addition of integers?
We will again use dots to illustrate the addition. Let a positive number be represented by a hollow dot and a negative number be represented by a solid dot.
A solid dot and a hollow dot are opposites and therefore when joined annul each other.

6. Ex: Consider the addition 3 + (-2)
We can illustrate the addition using solid and hollow dots = 3 + -2 1

7. Ex: Now consider the addition -3 + 2
Again illustrate the addition using solid and hollow dots = -3 + 2 -1

8. To recap: 3 + 2 = 5 same sign addends -3 + (-2) = -5
3 + (-2) = 1 different sign addends
= - 1
Can we describe a general rule for adding integers?
We see two cases: same sign addends
different sign addends

9. Addition of Integers When the addends have the same sign:
Add the absolute value of the addends. The sign of the sum will be the common sign of the addends.
When the addends have different signs:
Take the absolute value of the addends. Take the smaller from the larger absolute value. The sign of the sum will be same as the sign of the addend with larger absolute value.

10. Addition of Integers When the addends have the same sign:
Add the numbers and keep the sign.
When the addends have different signs:
Do a “take away” and keep the sign of the large “number”

11. We can also use the number line and direction arrows to illustrate addition of integers.
Let a positive number be represented by a right-facing arrow and a negative number be represented by a left-facing.
positive
negative

12. Ex: Model the addition 3 + 2 using the number line to find the sum.5
Start at zero and draw the first addend, 3
Positive 
From where the first arrow ends, draw the second addend, 2
Positive 
Where the second arrow ends is the sum

13. Start at zero and draw the first addend, -3 Negative 
Ex: Model the addition -3 + (-2) using the number line to find the sum. -5
Start at zero and draw the first addend, -3
Negative 
From where the first arrow ends, draw the second addend, -2
Negative 
Where the second arrow ends is the sum

14. Ex: Model the addition 3 + (-2) using the number line to find the sum.1
Start at zero and draw the first addend, 3
Positive 
From where the first arrow ends, draw the second addend, -2
Negative 
Where the second arrow ends is the sum

15. Ex: Model the addition -3 + 2 using the number line to find the sum.-1
Start at zero and draw the first addend, -3
Negative 
From where the first arrow ends, draw the second addend, 2
Positive 
Where the second arrow ends is the sum

16. Ex: Model the addition -5 + 3 using the number line to find the sum.-2
Start at zero and draw the first addend, -5
Negative 
From where the first arrow ends, draw the second addend, 3
Positive 
Where the second arrow ends is the sum

17. Ex: Model the addition problem 5 + (-3) to find the sum.= 5 + -3 2

18. Practice problems on handout

19. Subtraction of Integers
Ex: Consider the subtraction 3 – 2
Subtraction is defined to be adding the opposite.
The answer can be thought of as what is left when 2 is taken away from 3.
We can illustrate subtraction of integers using both dots and arrows, keeping in mind that subtraction is the opposite operation of addition.

20. Ex: Consider the subtraction 3 – 2 (take away)= 1 3 – 2
We want to take away 2 from the minuend
We conceptually understand the “taking-away” of like items to find the difference.

21. Ex: Consider the subtraction -3 – (-2) (take away)= -1 ––
We want to take away -2 from the minuend
We again conceptually understand the taking-away of like items to find the difference.

22. But in order to take away 3, I need 3 to begin with
But, what does 2 –– 3 mean? How can we illustrate subtraction of integers?
We will again use dots (solid and hollow) to illustrate the subtraction.
But in order to take away 3, I need 3 to begin with
 insert 1 solid and 1 hollow dot ( a “zero”)
Now take away 3 2 3 ––
And we are left with -1
take away
 2 – 3 = - 1

23. But in order to take away 3, I need 3 to begin with
Ex: Consider another take-away model to illustrate the subtraction 2 – 3. 2 3 ––
But in order to take away 3, I need 3 to begin with
 insert 3 solid and 3 hollow dots (which annul each other)
Now take away 3
We are left with -1
 2 – 3 = - 1

24. Now that we are adding, Just insert the 3 solid dots.
The previous take-away model can be simplified, we change subtraction to adding the opposite. 2 –– 3 
Now that we are adding, Just insert the 3 solid dots.
We are left with -1
 2 – 3 = - 1
 = - 1

25. Change subtraction to adding the opposite, insert 3 solid dots
Ex: Use the definition of subtraction to illustrate the subtraction -2 – 3. -2 3  ––
Change subtraction to adding the opposite,
insert 3 solid dots
We are left with -5
 -2 – 3  = - 5

26. Ex: Use the definition of subtraction to subtract: 2 – (-3)
 2 + (+3) ––
Just insert the 3 hollow dots (add the opposite of -3)
We are left with 5
 2 – (-3)  2 + (+3) = 5

27. Subtraction of Integers
Let a and b be integers.
Then a – b = a + (-b).
Change subtraction to addition and change the sign of what follows.

28. We can also use the number line and direction arrows to illustrate subtraction of integers. Let a positive number be represented by a right-facing arrow and a negative number be represented by a left-facing arrow.
positive
negative
The operation of subtraction acts to flip the direction of the number being subtracted’s arrow.

29. Start at zero and draw the first addend, 3 Positive 
Ex: Model the subtraction 3 – 2 using the number line to find the difference. 1
Start at zero and draw the first addend, 3
Positive 
From where the first arrow ends, draw the second addend, 2
Positive  
Remember, subtraction flips the arrow!
Where the second arrow ends is the difference

30. Start at zero and draw the first addend, 3 Positive 
Ex: Model the subtraction 3 – (-2) using the number line to find the difference. 5
Start at zero and draw the first addend, 3
Positive 
From where the first arrow ends, draw the second addend, - 2
Negative  
Remember, subtraction flips the arrow!
Where the second arrow ends is the difference

31. Start at zero and draw the first addend, -3 Negative 
Ex: Model the subtraction -3 – (-2) using the number line to find the difference. -1
Start at zero and draw the first addend, -3
Negative 
From where the first arrow ends, draw the second addend, - 2
Negative  
Remember, subtraction flips the arrow!
Where the second arrow ends is the difference

32. Start at zero and draw the first addend, -3 Negative 
Ex: Model the subtraction -3 – 2 using the number line to find the difference. -5
Start at zero and draw the first addend, -3
Negative 
From where the first arrow ends, draw the second addend, 2
Positive  
Remember, subtraction flips the arrow!
Where the second arrow ends is the difference

33. Practice problems on handout.

34. Multiplication of Integers
Ex: Consider the multiplication 3 x 2
The answer to the multiplication is how many three groups of 2 make (repeated addition).

35. Ex: Model the multiplication 3 x 2 using dots
3 x 2 represents three groups of 2: + +
3 x 2 = =
We conceptually understand the repeated addition of a positive number.

36. Ex: Model the multiplication 3 x (-2) using dots
3 x (-2) represents three groups of -2: (-2) + (-2) + +
3 x (-2) = =
We conceptually understand the repeated addition of a negative number.

37. But, what does -3 x 2 mean?
What does negative three groups of 2 represent?
The first factor is the repetition factor (how many times we are repeating the addition).
When that first factor is negative, we can think of repeated addition of the opposite of the second factor.

38. Ex: Model the multiplication -3 x 2 using dots
Negative repetition is repetition of the opposite of the second factor. + +
-3 x 2 = =

39. Ex: Model the multiplication -3 x (-2) using dots
-3 x (-2) represents negative three groups of -2
Negative repetition is repetition of the opposite + +
-3 x (-2) = =

40. To recap: 3 x 2 = 6 same sign factors -3 x (-2) = 6
-3 x 2 = -6 different sign factors
3 x (-2) = -6
Can we describe a general rule for multiplying integers?
We see two cases: same sign factors positive
different sign factors negative

41. Multiplication of Integers
Multiply and count the negative signs:
Even number of negative signs, result is positive,
Odd number of negative signs, result is negative

42. Practice problems - handout

43. Division of Integers Ex: Consider the division 6/3.
The answer to the division is if we partition the total number of items (6) into 3 groups, how many items are in each group?

44. Ex: Model the division 6/3 using the partition model.
Six divided by three: There are 6 dots (hollow).
Form 3 groups.
How many dots are in each group? 2
What kind of dots?
Solid  positive
 6/3 = 2

45. Ex: Model the division -6/3 using the partition model.
Negative Six divided by three: There are 6 dots (solid).
Form 3 groups.
How many dots are in each group? 2
What kind of dots?
Solid  negative
 -6/3 = -2

46. Ex: What does 6/(-3) mean?
Six divided by negative three: There are 6 dots (hollow).
Form -3 groups.
Huh?
The divisor represents the number of groups we will partition the dividend into.
* To negatively partition, we will partition the opposite.
Form 3 groups.
How many dots are in each group? 2
What kind of dots?
Solid  negative
 6/(-3) = -2

47. Ex: What does -6/(-3) mean?
Negative six divided by negative three: There are 6 dots (Solid).
Form -3 groups.
Huh?
* To negatively partition, we will partition the opposite.
Form 3 groups.
How many dots are in each group? 2
What kind of dots?
Solid  pos
 -6/(-3) = 2

48. To recap: 6/3 = 2 the same sign -6/(-3) = 2
-6/3 = different sign factors
6/(-3) = - 2
Can we describe a general rule for dividing integers?
We see two cases: same sign factors positive
different sign factors negative

49. Division of Integers When the dividend & divisor have the same sign:
Divide the absolute value of the factors The quotient will be positive.
When the dividend & divisors have different signs:
Divide the absolute value of the factors. The quotient will be negative.

50. Division of Integers Divide and count the negative signs:
Even number of negative signs, result is positive,
Odd number of negative signs, result is negative