Transcript 3 OpsIntegers wLines

OPERATIONS ON
INTEGERS
MSJC ~ San Jacinto Campus
Math Center Workshop Series
Janice Levasseur
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, . . .}
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.
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.
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.
Ex: Consider the addition 3 + (-2)
We can illustrate the addition using
solid and hollow dots
3
+
-2
=
1
Ex: Now consider the addition -3 + 2
Again illustrate the addition using solid and
hollow dots
-3
+
2
=
-1
To recap:
• 3+2=5
• -3 + (-2) = -5
same sign addends
• 3 + (-2) = 1
• -3 + 2 = - 1
different sign addends
Can we describe a general rule for adding integers?
We see two cases: same sign addends
different sign addends
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.
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”
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
Ex: Model the addition 3 + 2 using the number
line to find the sum.
0
5
Start at zero and draw the first addend, 3
Positive 
From where the first arrow ends, draw the
Positive 
second addend, 2
Where the second arrow ends is the sum
Ex: Model the addition -3 + (-2) using the
number line to find the sum.
-5
0
Start at zero and draw the first addend, -3
Negative 
From where the first arrow ends, draw the
Negative 
second addend, -2
Where the second arrow ends is the sum
Ex: Model the addition 3 + (-2) using the number
line to find the sum.
0 1
Start at zero and draw the first addend, 3
Positive 
From where the first arrow ends, draw the
Negative 
second addend, -2
Where the second arrow ends is the sum
Ex: Model the addition -3 + 2 using the number
line to find the sum.
-1 0
Start at zero and draw the first addend, -3
Negative 
From where the first arrow ends, draw the
Positive 
second addend, 2
Where the second arrow ends is the sum
Ex: Model the addition -5 + 3 using the number
line to find the sum.
-2
0
Start at zero and draw the first addend, -5
Negative 
From where the first arrow ends, draw the
Positive 
second addend, 3
Where the second arrow ends is the sum
Ex: Model the addition problem 5 + (-3)
to find the sum.
5
+
-3
=
2
Practice problems on handout
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.
Ex: Consider the subtraction 3 – 2 (take away)
3
–
2
=
1
We want to take away 2 from the minuend
We conceptually understand the “taking-away”
of like items to find the difference.
Ex: Consider the subtraction -3 – (-2) (take away)
-3
––
-2
=
-1
We want to take away -2 from the minuend
We again conceptually understand the takingaway of like items to find the difference.
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”)
2
––
3
take away
Now take away 3
And we are left with -1
2–3=-1
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
2–3=-1
-1
The previous take-away model can be simplified,
we change subtraction to adding the opposite.
2
––
3
 2 + -3
Now that we are adding,
Just insert the 3 solid dots.
We are left with -1
 2 – 3 = - 1  2 + -3 = - 1
Ex: Use the definition of subtraction to illustrate the
subtraction -2 – 3.
-2
––
3  -2 + -3
Change subtraction to
adding the opposite,
insert 3 solid dots
We are left with -5
 -2 – 3  -2 + -3 = - 5
Ex: Use the definition of subtraction to subtract:
2 – (-3)
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
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.
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.
Ex: Model the subtraction 3 – 2 using the
number line to find the difference.
0 1
Start at zero and draw the first addend, 3
Positive 
From where the first arrow ends, draw the
Positive 

second addend, 2
Remember, subtraction flips the arrow!
Where the second arrow ends is the difference
Ex: Model the subtraction 3 – (-2) using the
number line to find the difference.
0
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
Ex: Model the subtraction -3 – (-2) using the
number line to find the difference.
-1 0
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
Ex: Model the subtraction -3 – 2 using the
number line to find the difference.
-5
0
Start at zero and draw the first addend, -3
Negative 
From where the first arrow ends, draw the
Positive 

second addend, 2
Remember, subtraction flips the arrow!
Where the second arrow ends is the difference
Practice problems on handout.
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).
Ex: Model the multiplication 3 x 2 using dots
3 x 2 represents three groups of 2: 2 + 2 + 2
+
3x2=
2
+
+
2
+
2
=
6
We conceptually understand the repeated addition
of a positive number.
Ex: Model the multiplication 3 x (-2) using dots
3 x (-2) represents three groups of -2: -2 + (-2) + (-2)
+
3 x (-2) = -2
+
+
-2
+
-2
=
-6
We conceptually understand the repeated addition
of a negative number.
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.
Ex: Model the multiplication -3 x 2 using dots
Negative repetition is repetition of the opposite of the
second factor.
+
-3 x 2 = -2
+
+
-2
+
-2
=
-6
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) = 2
+
+
2
+
2
=
6
To recap:
• 3x2=6
• -3 x (-2) = 6
same sign factors
• -3 x 2 = -6
• 3 x (-2) = -6
different sign factors
Can we describe a general rule for
multiplying integers?
We see two cases: same sign factors
different sign factors
positive
negative
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
Practice problems - handout
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?
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
2
group?
What kind of dots?
Solid  positive
 6/3 = 2
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
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.
Form 3 groups.
* To negatively partition,
we will partition the opposite.
How many dots are in each group? 2
What kind of dots? Solid  negative  6/(-3) = -2
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
To recap:
• 6/3 = 2
• -6/(-3) = 2
the same sign
• -6/3 = - 2
• 6/(-3) = - 2
different sign factors
Can we describe a general rule for dividing integers?
We see two cases: same sign factors
positive
different sign factors negative
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.
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