OPERATIONS ON INTEGERS

MSJC ~ San Jacinto CampusMath 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 IntegersEx: 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 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 IntegersWhen 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 IntegersWhen 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”

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

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.

Subtraction is defined to be adding the opposite.

Ex: Consider the subtraction 3 – 2 (take away)

3 – 2 =

We want to take away 2 from the minuend

1

We conceptually understand the “taking-away” of like items to find the difference.

Ex: Consider the subtraction -3 – (-2) (take away)

-3 –– -2 =

We want to take away -2 from the minuend

-1

We again conceptually understand the taking-away 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.

2 ––

take away

3

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

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 -1

2 – 3 = - 1

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

2 + -3

2 + -3 = - 1

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 -2 + -3 = - 5

-2 + -3

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

2 + (+3)

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.

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

+ +

3 x 2 = 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

+ +

-3 x 2 = -2 + -2 + -2 = -6

Negative repetition is repetition of the opposite of the second factor.

Ex: Model the multiplication -3 x (-2) using dots

-3 x (-2) represents negative three groups of -2

+ +

-3 x (-2) = 2 + 2 + 2 = 6

Negative repetition is repetition of the opposite

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 IntegersMultiply and count the negative signs:

Even number of negative signs, result is positive,

Odd number of negative signs, result is negative

Practice problems - handout

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?

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

6/3 = 2

What kind of dots?

Hollow positive

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

-6/3 = -2

What kind of dots?

Solid negative

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

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

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 IntegersWhen 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.