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§ 5-1 Addition and Subtraction of Integers

Integers

What are integers? How are they related to whole numbers?

DefinitionAn integer is a number that can be written without a fractionalcomponent.

In this section we will focus on addition and subtraction of integers.Let’s start with the properties under addition. So how do we defineaddition of integers?

It isn’t as easy as it sounds ...

Integers

Definition of Addition

To define addition of integers, we need to consider cases:

The Addition of IntegersLet a, b ∈ Z.

0 + a = a = a + 0

If a ≥ 0 and b ≥ 0 then a + b is defined as we did for wholenumbers.

If a ≥ 0 and b ≥ 0 then −a + (−b) = −(a + b).

If b > 0 and a ≥ b then a + (−b) = a− b.

If b > a and a > 0 then a + (−b) = −(b− a).

0 + a = a = a + 0

Addition Properties

Are the integers closed under addition?

Closure of the Integers Under AdditionIf a, b ∈ Z, then a + b is a unique integer.

Does the commutative property hold with respect to addition?

Commutative Property of Addition of IntegersIf a, b ∈ Z, then a + b = b + a

Does the associative property hold with respect to addition?

Associative Property of Addition of Integers

If a, b, c ∈ Z, then a + (b + c) = (a + b) + c

Addition Properties

Is there a unique identity element?

Identity Property of Addition of IntegersFor every integer a, 0 is the unique additive identity such thata + 0 = 0 + a = a.

Is there an element in the integers that adds to a given integer toarrive at the identity?

Inverse Property of Addition of IntegersFor every integer a, there exists the unique integer −a such thata + (−a) = −a + a = 0.

Addition Properties

Representing Integer Addition

We can use number lines, but now we need to consider the negativeside.

ExampleIllustrate 3 + 2.

-5 -4 -3 -2 -1 0 1 2 3 4 5

3 + 2 = 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

3 + 2 = 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

3 + 2 = 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

3 + 2 = 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

3 + 2 = 5

-5 -4 -3 -2 -1 0 1 2 3 4 5

3 + 2 = 5

Example

Illustrate 3 + (−2).

-5 -4 -3 -2 -1 0 1 2 3 4 5

Example

-5 -4 -3 -2 -1 0 1 2 3 4 5

Example

-5 -4 -3 -2 -1 0 1 2 3 4 5

Example

-5 -4 -3 -2 -1 0 1 2 3 4 5

Example

-5 -4 -3 -2 -1 0 1 2 3 4 5

ExampleIllustrate −2 + 3.

-5 -4 -3 -2 -1 0 1 2 3 4 5

Example

Illustrate −2 + (−3).

-5 -4 -3 -2 -1 0 1 2 3 4 5

Example

-5 -4 -3 -2 -1 0 1 2 3 4 5

Example

-5 -4 -3 -2 -1 0 1 2 3 4 5

Example

-5 -4 -3 -2 -1 0 1 2 3 4 5

Addition Properties

More Properties of Integer AdditionLet a, b, c be integers.

−(−a) = a

If a = b then a + c = b + c

−a + (−b) = −(a + b)

Addition Properties

−(−a) = a

−a + (−b) = −(a + b)

Addition Properties

−(−a) = a

−a + (−b) = −(a + b)

Another common method is using colored chips.

Example

Represent 5 + (−6) using chips.

5 + (−6) = −1

Example

5 + (−6) = −1

Example

5 + (−6) = −1

Example

5 + (−6) = −1

Example

5 + (−6) = −1

Example

5 + (−6) = −1

We can augment this a bit to use the charged field method.

Example

Represent 5 + (−6) using the charged field method.

− − −− − −

5 + (−6) = −1

Example

− − −− − −

5 + (−6) = −1

Example

− − −− − −

5 + (−6) = −1

Example

− − −− − −

5 + (−6) = −1

Example

− − −− − −

5 + (−6) = −1

Example

− − −− − −

5 + (−6) = −1

Subtraction of Integers

Let’s start with a definition in terms of addition of integers.

DefinitionIf a, b ∈ Z, a− b is the unique integer n such that a = b + n.

TheoremFor all integers a and b, a− b is the same as a + (−b).

Properties of Subtraction of Integers

Are the integers closed under subtraction?

Closure of the Integers Under SubtractionIf a, b ∈ Z, then a− b is a unique integer.

Do the integers commute with respect to subtraction?

Are the integers associative with respect to subtraction?

Is there a unique identity element for subtraction of integers?

Representing Subtraction

The idea for subtraction is this: for subtraction of positive integers, itis exactly the same as addition. But when subtracting negativeintegers, we have to take into account the double negatives.

Think of it this way: when subtracting positive integers, what does thenegative sign tell us to do? turn around

We use this idea when subtracting negative integers too. We just haveto also take into account the negative with the integer. That is, afterwe turn around, the negative tells us to move ‘backwards’.

Think of it this way: when subtracting positive integers, what does thenegative sign tell us to do?

turn around

Example

Illustrate −2− (−5).

-5 -4 -3 -2 -1 0 1 2 3 4 5

Example

-5 -4 -3 -2 -1 0 1 2 3 4 5

Example

-5 -4 -3 -2 -1 0 1 2 3 4 5

Example

-5 -4 -3 -2 -1 0 1 2 3 4 5

Example

-5 -4 -3 -2 -1 0 1 2 3 4 5

We can also use the chip method, but we have to be creative here aswell. The idea is that we cannot take negatives from positives, so weneed to create a way to have negatives in our picture.

Example

Illustrate 2− (−1).

Example

Absolute Value

We all know what absolute value represents, right?

To define absolute value, however, we need negative integers.

DefinitionThe absolute value of an integer a is given by

|a| ={

a a ≥ 0−a a < 0

In other words, the absolute value represents the distance the numberlies from 0 on the number line.

Absolute Value

|a| ={

a a ≥ 0−a a < 0

Absolute Value

|a| ={

a a ≥ 0−a a < 0

Absolute Value

|a| ={

a a ≥ 0−a a < 0

Absolute Value

ExampleFind the following:

| − 2| = −(−2) = 2

| − 3 + 2| = | − 1| = −(−1) = 1

| − 2− 5| = | − 7| = −(−7) = 7

−|3− 5| = −| − 2| = −(−(−2)) = −2

Find x such that |x− 2| = 5x = 7,−3

Absolute Value

|3| = 3

| − 2| = −(−2) = 2

| − 3 + 2| = | − 1| = −(−1) = 1

| − 2− 5| = | − 7| = −(−7) = 7

−|3− 5| = −| − 2| = −(−(−2)) = −2

Absolute Value

|3| = 3

| − 2|

= −(−2) = 2

| − 3 + 2| = | − 1| = −(−1) = 1

| − 2− 5| = | − 7| = −(−7) = 7

−|3− 5| = −| − 2| = −(−(−2)) = −2

Absolute Value

|3| = 3

| − 2| = −(−2) = 2

| − 3 + 2| = | − 1| = −(−1) = 1

| − 2− 5| = | − 7| = −(−7) = 7

−|3− 5| = −| − 2| = −(−(−2)) = −2

Absolute Value

|3| = 3

| − 2| = −(−2) = 2

| − 3 + 2|

= | − 1| = −(−1) = 1

| − 2− 5| = | − 7| = −(−7) = 7

−|3− 5| = −| − 2| = −(−(−2)) = −2

Absolute Value

|3| = 3

| − 2| = −(−2) = 2

| − 3 + 2| = | − 1| = −(−1) = 1

| − 2− 5| = | − 7| = −(−7) = 7

−|3− 5| = −| − 2| = −(−(−2)) = −2

Absolute Value

|3| = 3

| − 2| = −(−2) = 2

| − 3 + 2| = | − 1| = −(−1) = 1

| − 2− 5|

= | − 7| = −(−7) = 7

−|3− 5| = −| − 2| = −(−(−2)) = −2

Absolute Value

|3| = 3

| − 2| = −(−2) = 2

| − 3 + 2| = | − 1| = −(−1) = 1

| − 2− 5| = | − 7| = −(−7) = 7

−|3− 5| = −| − 2| = −(−(−2)) = −2

Absolute Value

|3| = 3

| − 2| = −(−2) = 2

| − 3 + 2| = | − 1| = −(−1) = 1

| − 2− 5| = | − 7| = −(−7) = 7

−|3− 5|

= −| − 2| = −(−(−2)) = −2

Absolute Value

|3| = 3

| − 2| = −(−2) = 2

| − 3 + 2| = | − 1| = −(−1) = 1

| − 2− 5| = | − 7| = −(−7) = 7

−|3− 5| = −| − 2| = −(−(−2)) = −2

Absolute Value

|3| = 3

| − 2| = −(−2) = 2

| − 3 + 2| = | − 1| = −(−1) = 1

| − 2− 5| = | − 7| = −(−7) = 7

−|3− 5| = −| − 2| = −(−(−2)) = −2

Find x such that |x− 2| = 5

x = 7,−3

Absolute Value

|3| = 3

| − 2| = −(−2) = 2

| − 3 + 2| = | − 1| = −(−1) = 1

| − 2− 5| = | − 7| = −(−7) = 7

−|3− 5| = −| − 2| = −(−(−2)) = −2

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