OPERATIONS ON INTEGERS

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OPERATIONS ON INTEGERS

• MSJC San Jacinto Campus
• Math Center Workshop Series
• Janice Levasseur

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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,
  . . .

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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.
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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.
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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.
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Ex Consider the addition 3 (-2)
We can illustrate the addition using solid and
hollow dots

3

-2
1
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Ex Now consider the addition -3 2
Again illustrate the addition using solid and
hollow dots

-3

2
-1
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To recap

• 3 2 5 same sign addends
• -3 (-2) -5
• 3 (-2) 1 different sign addends
• -3 2 - 1

Can we describe a general rule for adding
integers?
We see two cases same sign addends
different sign addends
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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.
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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
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Ex Model the addition problem 5 (-3) to find
the sum.

5

-3
2
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Practice problems on handout
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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.
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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.
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Ex Consider the subtraction -3 (-2) (take
away)

-1
-3
-2

We want to take away -2 from the minuend
We again conceptually understand the taking-away
of like items to find the difference.
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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
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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
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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
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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
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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
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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.
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Practice problems on handout.
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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).
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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
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We conceptually understand the repeated addition
of a positive number.
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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.
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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.
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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
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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
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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
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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
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Practice problems - handout
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Practice problems - handout
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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?
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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?
Hollow ? positive
? 6/3 2
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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
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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
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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?
Hollow ? positive
? -6/(-3) 2
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To recap

• 6/3 2 the same sign
• -6/(-3) 2
• -6/3 - 2 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
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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.
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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.