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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, . . .}

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operations on integers

OPERATIONS ON INTEGERS

MSJC ~ San Jacinto Campus

Math Center Workshop Series

Janice Levasseur

basic definitions
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
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.

slide4

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.

slide5

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.

slide6

Ex: Consider the addition 3 + (-2)

We can illustrate the addition using solid and hollow dots

=

3

+

-2

1

slide7

Ex: Now consider the addition -3 + 2

Again illustrate the addition using solid and hollow dots

=

-3

+

2

-1

to recap
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

addition of integers1
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 integers2
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”

subtraction of integers
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.

slide14

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.

slide15

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.

slide16

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

slide17

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

slide18

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

slide19

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

slide20

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 integers1
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.

multiplication of integers
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).

slide24

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 positive number.

slide25

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.

slide26

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.

slide27

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

slide28

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 recap1
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

multiplication of integers1
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

division of integers
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?

slide34

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

slide35

Ex: Model the division -6/3 using the partition model.

Negative Sixdivided 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

slide36

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

slide37

Ex: What does -6/(-3) mean?

Negative sixdivided 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

to recap2
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

division of integers1
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 integers2
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.

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