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Properties of Addition and Multiplication
Joseph Lee
Metropolitan Community College
Joseph Lee
Properties of Addition and Multiplication
Goals
In this section, we introduce properties that will be essential to
performing algebra for the remainder of the course.
These properties include:
Joseph Lee
Properties of Addition and Multiplication
Goals
In this section, we introduce properties that will be essential to
performing algebra for the remainder of the course.
These properties include:
1
the commutative property
Joseph Lee
Properties of Addition and Multiplication
Goals
In this section, we introduce properties that will be essential to
performing algebra for the remainder of the course.
These properties include:
1
the commutative property
2
the associative property
Joseph Lee
Properties of Addition and Multiplication
Goals
In this section, we introduce properties that will be essential to
performing algebra for the remainder of the course.
These properties include:
1
the commutative property
2
the associative property
3
the distributive property
Joseph Lee
Properties of Addition and Multiplication
Goals
In this section, we introduce properties that will be essential to
performing algebra for the remainder of the course.
These properties include:
1
the commutative property
2
the associative property
3
the distributive property
4
the identity property
Joseph Lee
Properties of Addition and Multiplication
Goals
In this section, we introduce properties that will be essential to
performing algebra for the remainder of the course.
These properties include:
1
the commutative property
2
the associative property
3
the distributive property
4
the identity property
5
the inverse property
Joseph Lee
Properties of Addition and Multiplication
Definition: Commutative Property
The commutative property of addition states that for any real
numbers a and b,
a + b = b + a.
In other words, it doesn’t matter in which order you add.
Joseph Lee
Properties of Addition and Multiplication
Definition: Commutative Property
The commutative property of addition states that for any real
numbers a and b,
a + b = b + a.
In other words, it doesn’t matter in which order you add.
The commutative property of multiplication states that for any real
numbers a and b,
a · b = b · a.
In other words, it doesn’t matter in which order you multiply.
Joseph Lee
Properties of Addition and Multiplication
Definition: Associative Property
The associative property of addition states that for any real
numbers a, b, and c,
a + (b + c) = (a + b) + c
In other words, it doesn’t matter how you group when you add.
Joseph Lee
Properties of Addition and Multiplication
Definition: Associative Property
The associative property of addition states that for any real
numbers a, b, and c,
a + (b + c) = (a + b) + c
In other words, it doesn’t matter how you group when you add.
The associative property of multiplication states that for any real
numbers a and b,
a · (b · c) = (a · b) · c.
In other words, it doesn’t matter how you group when you multiply.
Joseph Lee
Properties of Addition and Multiplication
Definition: Distributive Property
The distributive property (of multiplication over addition) states
that for any real numbers a, b, and c,
a(b + c) = ab + ac
Joseph Lee
Properties of Addition and Multiplication
Definition: Distributive Property
The distributive property (of multiplication over addition) states
that for any real numbers a, b, and c,
a(b + c) = ab + ac
This property is the fundamental connection between addition and
multiplication.
Joseph Lee
Properties of Addition and Multiplication
Definition: Distributive Property
The distributive property (of multiplication over addition) states
that for any real numbers a, b, and c,
a(b + c) = ab + ac
This property is the fundamental connection between addition and
multiplication.
Note:
3(2) = 3(1 + 1) = 3 + 3 = 6
Joseph Lee
Properties of Addition and Multiplication
Definition: Distributive Property
The distributive property (of multiplication over addition) states
that for any real numbers a, b, and c,
a(b + c) = ab + ac
This property is the fundamental connection between addition and
multiplication.
Note:
3(2) = 3(1 + 1) = 3 + 3 = 6
or
3(4) = 3(1 + 1 + 1 + 1) = 3 + 3 + 3 + 3 = 12
Joseph Lee
Properties of Addition and Multiplication
Definition: Identity Property
The identity property of addition states that for any real number a,
you may always add the identity element of addition, 0, and their
sum will be a, i.e.
a+0=a
Joseph Lee
Properties of Addition and Multiplication
Definition: Identity Property
The identity property of addition states that for any real number a,
you may always add the identity element of addition, 0, and their
sum will be a, i.e.
a+0=a
The identity property of multiplication states that for any real
number a, you may always add the identity element of
multiplication, 1, and their product will be a, i.e.
a·1=a
Joseph Lee
Properties of Addition and Multiplication
Definition: Inverse Property
The inverse property of addition states that for any real number a,
there is always another real number −a such that their sum is the
identity element for addition, i.e.
a + (−a) = 0
Joseph Lee
Properties of Addition and Multiplication
Definition: Inverse Property
The inverse property of addition states that for any real number a,
there is always another real number −a such that their sum is the
identity element for addition, i.e.
a + (−a) = 0
The inverse property of multiplication states that for any real
number a (except 0), there is always another real number 1a such
that their sum is the identity element for multiplication, i.e.
a·
Joseph Lee
1
=1
a
Properties of Addition and Multiplication
Example 1.a
Identify the property demonstrated.
(x + 3) + 6 = x + (3 + 6)
Joseph Lee
Properties of Addition and Multiplication
Example 1.a
Identify the property demonstrated.
(x + 3) + 6 = x + (3 + 6)
Solution. Associative Property of Addition
Joseph Lee
Properties of Addition and Multiplication
Example 1.b
Identify the property demonstrated.
7(x + 3) = 7x + 21
Joseph Lee
Properties of Addition and Multiplication
Example 1.b
Identify the property demonstrated.
7(x + 3) = 7x + 21
Solution. Distributive Property
Joseph Lee
Properties of Addition and Multiplication
Example 1.c
Identify the property demonstrated.
4·
Joseph Lee
1
=1
4
Properties of Addition and Multiplication
Example 1.c
Identify the property demonstrated.
4·
1
=1
4
Solution. Inverse Property of Multiplication
Joseph Lee
Properties of Addition and Multiplication
Example 1.d
Identify the property demonstrated.
3(7x) = (7x)(3)
Joseph Lee
Properties of Addition and Multiplication
Example 1.d
Identify the property demonstrated.
3(7x) = (7x)(3)
Solution. Commutative Property of Multiplication
Joseph Lee
Properties of Addition and Multiplication
Example 1.e
Identify the property demonstrated.
4x 2 + 0 = 4x 2
Joseph Lee
Properties of Addition and Multiplication
Example 1.e
Identify the property demonstrated.
4x 2 + 0 = 4x 2
Solution. Identity Property of Addition
Joseph Lee
Properties of Addition and Multiplication
Example 1.f
Identify the property demonstrated.
(3x + 2) + 4x = 4x + (3x + 2)
Joseph Lee
Properties of Addition and Multiplication
Example 1.f
Identify the property demonstrated.
(3x + 2) + 4x = 4x + (3x + 2)
Solution. Commutative Property of Addition
Joseph Lee
Properties of Addition and Multiplication
Example 1.g
Identify the property demonstrated.
3x + (−3x) = 0
Joseph Lee
Properties of Addition and Multiplication
Example 1.g
Identify the property demonstrated.
3x + (−3x) = 0
Solution. Inverse Property of Addition
Joseph Lee
Properties of Addition and Multiplication
Example 1.h
Identify the property demonstrated.
4(5x − 2) = (5x − 2)(4)
Joseph Lee
Properties of Addition and Multiplication
Example 1.h
Identify the property demonstrated.
4(5x − 2) = (5x − 2)(4)
Solution. Commutative Property of Multiplication
Joseph Lee
Properties of Addition and Multiplication
Example 1.i
Identify the property demonstrated.
4(5x − 2) = 20x − 8
Joseph Lee
Properties of Addition and Multiplication
Example 1.i
Identify the property demonstrated.
4(5x − 2) = 20x − 8
Solution. Distributive Property
Joseph Lee
Properties of Addition and Multiplication
Example 1.j
Identify the property demonstrated.
1
1
·1=
x
x
Joseph Lee
Properties of Addition and Multiplication
Example 1.j
Identify the property demonstrated.
1
1
·1=
x
x
Solution. Identity Property of Multiplication
Joseph Lee
Properties of Addition and Multiplication
Example 2.a
Apply the commutative property of addition to the given
expression.
8x + 4
Joseph Lee
Properties of Addition and Multiplication
Example 2.a
Apply the commutative property of addition to the given
expression.
8x + 4
Solution.
8x + 4 =
Joseph Lee
Properties of Addition and Multiplication
Example 2.a
Apply the commutative property of addition to the given
expression.
8x + 4
Solution.
8x + 4 = 4 + 8x
Joseph Lee
Properties of Addition and Multiplication
Example 2.b
Apply the associative property of multiplication to the given
expression.
2(4x)
Joseph Lee
Properties of Addition and Multiplication
Example 2.b
Apply the associative property of multiplication to the given
expression.
2(4x)
Solution.
2(4x) =
Joseph Lee
Properties of Addition and Multiplication
Example 2.b
Apply the associative property of multiplication to the given
expression.
2(4x)
Solution.
2(4x) = (2 · 4)x
Joseph Lee
Properties of Addition and Multiplication
Example 2.c
Apply the distributive property to the given expression.
3(4x − 9)
Joseph Lee
Properties of Addition and Multiplication
Example 2.c
Apply the distributive property to the given expression.
3(4x − 9)
Solution.
3(4x − 9) =
Joseph Lee
Properties of Addition and Multiplication
Example 2.c
Apply the distributive property to the given expression.
3(4x − 9)
Solution.
3(4x − 9) = 12x − 27
Joseph Lee
Properties of Addition and Multiplication