Post on 03-Aug-2020
§ 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
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
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
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 ...
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).
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).
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).
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).
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).
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
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
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
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
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
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
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
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
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.
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
3 + 2 = 5
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
3 + 2 = 5
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
3 + 2 = 5
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
3 + 2 = 5
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
3 + 2 = 5
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
3 + 2 = 5
Representing Integer Addition
Example
Illustrate 3 + (−2).
-5 -4 -3 -2 -1 0 1 2 3 4 5
3
−2
Representing Integer Addition
Example
Illustrate 3 + (−2).
-5 -4 -3 -2 -1 0 1 2 3 4 5
3
−2
Representing Integer Addition
Example
Illustrate 3 + (−2).
-5 -4 -3 -2 -1 0 1 2 3 4 5
3
−2
Representing Integer Addition
Example
Illustrate 3 + (−2).
-5 -4 -3 -2 -1 0 1 2 3 4 5
3
−2
Representing Integer Addition
Example
Illustrate 3 + (−2).
-5 -4 -3 -2 -1 0 1 2 3 4 5
3
−2
Representing Integer Addition
ExampleIllustrate −2 + 3.
-5 -4 -3 -2 -1 0 1 2 3 4 5
−2
3
Representing Integer Addition
ExampleIllustrate −2 + 3.
-5 -4 -3 -2 -1 0 1 2 3 4 5
−2
3
Representing Integer Addition
ExampleIllustrate −2 + 3.
-5 -4 -3 -2 -1 0 1 2 3 4 5
−2
3
Representing Integer Addition
ExampleIllustrate −2 + 3.
-5 -4 -3 -2 -1 0 1 2 3 4 5
−2
3
Representing Integer Addition
Example
Illustrate −2 + (−3).
-5 -4 -3 -2 -1 0 1 2 3 4 5
−2
3
Representing Integer Addition
Example
Illustrate −2 + (−3).
-5 -4 -3 -2 -1 0 1 2 3 4 5
−2
3
Representing Integer Addition
Example
Illustrate −2 + (−3).
-5 -4 -3 -2 -1 0 1 2 3 4 5
−2
3
Representing Integer Addition
Example
Illustrate −2 + (−3).
-5 -4 -3 -2 -1 0 1 2 3 4 5
−2
3
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
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
More Properties of Integer AdditionLet a, b, c be integers.
−(−a) = a
If a = b then a + c = b + c
−a + (−b) = −(a + b)
Representing Integer Addition
Another common method is using colored chips.
Example
Represent 5 + (−6) using chips.
5 + (−6) = −1
Representing Integer Addition
Another common method is using colored chips.
Example
Represent 5 + (−6) using chips.
5 + (−6) = −1
Representing Integer Addition
Another common method is using colored chips.
Example
Represent 5 + (−6) using chips.
5 + (−6) = −1
Representing Integer Addition
Another common method is using colored chips.
Example
Represent 5 + (−6) using chips.
5 + (−6) = −1
Representing Integer Addition
Another common method is using colored chips.
Example
Represent 5 + (−6) using chips.
5 + (−6) = −1
Representing Integer Addition
Another common method is using colored chips.
Example
Represent 5 + (−6) using chips.
5 + (−6) = −1
Representing Integer Addition
We can augment this a bit to use the charged field method.
Example
Represent 5 + (−6) using the charged field method.
++
++
+
− − −− − −
5 + (−6) = −1
Representing Integer Addition
We can augment this a bit to use the charged field method.
Example
Represent 5 + (−6) using the charged field method.
++
++
+
− − −− − −
5 + (−6) = −1
Representing Integer Addition
We can augment this a bit to use the charged field method.
Example
Represent 5 + (−6) using the charged field method.
++
++
+
− − −− − −
5 + (−6) = −1
Representing Integer Addition
We can augment this a bit to use the charged field method.
Example
Represent 5 + (−6) using the charged field method.
++
++
+
− − −− − −
5 + (−6) = −1
Representing Integer Addition
We can augment this a bit to use the charged field method.
Example
Represent 5 + (−6) using the charged field method.
++
++
+
− − −− − −
5 + (−6) = −1
Representing Integer Addition
We can augment this a bit to use the charged field method.
Example
Represent 5 + (−6) using the charged field method.
++
++
+
− − −− − −
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).
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).
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?
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?
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?
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?
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’.
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’.
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’.
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’.
Representing Subtraction
Example
Illustrate −2− (−5).
-5 -4 -3 -2 -1 0 1 2 3 4 5
−2
−5
Representing Subtraction
Example
Illustrate −2− (−5).
-5 -4 -3 -2 -1 0 1 2 3 4 5
−2
−5
Representing Subtraction
Example
Illustrate −2− (−5).
-5 -4 -3 -2 -1 0 1 2 3 4 5
−2
−5
Representing Subtraction
Example
Illustrate −2− (−5).
-5 -4 -3 -2 -1 0 1 2 3 4 5
−2
−5
Representing Subtraction
Example
Illustrate −2− (−5).
-5 -4 -3 -2 -1 0 1 2 3 4 5
−2
−5
Representing Subtraction
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).
Representing Subtraction
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).
Representing Subtraction
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).
Representing Subtraction
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).
Representing Subtraction
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).
Representing Subtraction
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).
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
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
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
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
ExampleFind the following:
|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| = 5x = 7,−3
Absolute Value
ExampleFind the following:
|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| = 5x = 7,−3
Absolute Value
ExampleFind the following:
|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| = 5x = 7,−3
Absolute Value
ExampleFind the following:
|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| = 5x = 7,−3
Absolute Value
ExampleFind the following:
|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| = 5x = 7,−3
Absolute Value
ExampleFind the following:
|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| = 5x = 7,−3
Absolute Value
ExampleFind the following:
|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| = 5x = 7,−3
Absolute Value
ExampleFind the following:
|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| = 5x = 7,−3
Absolute Value
ExampleFind the following:
|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| = 5x = 7,−3
Absolute Value
ExampleFind the following:
|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| = 5x = 7,−3
Absolute Value
ExampleFind the following:
|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
ExampleFind the following:
|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| = 5x = 7,−3