Download Properties of Real Numbers - Lakewood City School District

Properties of Real Numbers
Commutative
Associative
Distributive
Identity + ×
Inverse + ×
Commutative Properties
Changing the order of the numbers in
addition or multiplication will not
change the result.
Commutative Property of Addition
states: 2 + 3 = 3 + 2 or a + b = b + a.
Commutative Property of
Multiplication states: 4 • 5 = 5 • 4 or
ab = ba.
Associative Properties
Changing the grouping of the
numbers in addition or multiplication
will not change the result.
Associative Property of Addition
states: 3 + (4 + 5)= (3 + 4)+ 5 or
a + (b + c)= (a + b)+ c
Associative Property of Multiplication
states: (2 • 3) • 4 = 2 • (3 • 4) or
(ab)c = a(bc)
Distributive Property
Multiplication distributes over addition.
ab  c   ab  ac
32  5  3  2  3  5
Additive Identity Property
There exists a unique number 0 such
that zero preserves identities under
addition.
a + 0 = a and 0 + a = a
In other words adding zero to a
number does not change its value.
Multiplicative Identity Property
There exists a unique number 1 such
that the number 1 preserves identities
under multiplication.
a ∙ 1 = a and 1 ∙ a = a
In other words multiplying a number
by 1 does not change the value of the
number.
Additive Inverse Property
For each real number a there
exists a unique real number –a
such that their sum is zero.
a + (-a) = 0
In other words opposites add to
zero.
Multiplicative Inverse Property
For each real number a there exists a
unique real number
product is 1.
1
a 1
a
1
a
such that their
Let’s play “Name that property!”
State the property or properties that
justify the following.
3+2=2+3
State the property or properties that
justify the following.
3+2=2+3
Commutative Property
State the property or properties that
justify the following.
10(1/10) = 1
State the property or properties that
justify the following.
10(1/10) = 1
Multiplicative Inverse Property
State the property or properties that
justify the following.
3(x – 10) = 3x – 30
State the property or properties that
justify the following.
3(x – 10) = 3x – 30
Distributive Property
State the property or properties that
justify the following.
3 + (4 + 5) = (3 + 4) + 5
State the property or properties that
justify the following.
3 + (4 + 5) = (3 + 4) + 5
Associative Property
State the property or properties that
justify the following.
(5 + 2) + 9 = (2 + 5) + 9
State the property or properties that
justify the following.
(5 + 2) + 9 = (2 + 5) + 9
Commutative Property
3+7=7+3
3+7=7+3
Commutative
Property of Addition
8+0=8
8+0=8
Identity Property of
Addition
6•4=4•6
6•4=4•6
Commutative Property
of Multiplication
17 + (-17) = 0
17 + (-17) = 0
Inverse Property of
Addition
2(5) = 5(2)
2(5) = 5(2)
Commutative Property
of Multiplication
(2 + 1) + 4 = 2 + (1 + 4)
(2 + 1) + 4 = 2 + (1 + 4)
Associative Property
of Addition
even + even = even
3(2 + 5) = 3•2 + 3•5
3(2 + 5) = 3•2 + 3•5
Distributive Property
6(7•8) = (6•7)8
6(7•8) = (6•7)8
Associative Property of
Multiplication
5•1=5
5•1=5
Identity Property of
Multiplication
(6 – 3)4 = 6•4 – 3•4
(6 – 3)4 = 6•4 – 3•4
Distributive Property
1(-9) = -9
1(-9) = -9
Identity Property of
Multiplication
3 + (-3) = 0
3 + (-3) = 0
Inverse Property of
Addition
1 + [-9 + 3]
= [1 + (-9)] + 3
1 + [-9 + 3]
= [1 + (-9)] + 3
Associative Property of
Addition
-3(6) = 6(-3)
-3(6) = 6(-3)
Commutative Property
of Multiplication
-8 + 0 = -8
-8 + 0 = -8
Identity Property of
Addition
3•7 – 3•4 = 3(7 – 4)
3•7 – 3•4 = 3(7 – 4)
Distributive Property
6 + [(3 + (-2)]
= (6 + 3) + (- 2)
6 + [(3 + (-2)]
= (6 + 3) + (- 2)
Associative Property
of Addition
7 + (-5) = -5 + 7
7 + (-5) = -5 + 7
Commutative
Property of Addition
(5 + 4)9 = 45 + 36
(5 + 4)9 = 45 + 36
Distributive Property
-3(5 • 4) = (-3 • 5)4
-3(5 • 4) = (-3 • 5)4
Associative Property of
Multiplication
-8(4) = 4(-8)
-8(4) = 4(-8)
Commutative Property
of Multiplication
1
5/
+
0
=
7
1
5/
7
1
5/
+
0
=
7
1
5/
7
Identity Property of
Addition
3/
–
4
6/
=
–
7
6/
+
7
3/
4
3/
–
4
6/
=
–
7
6/
+
7
3/
4
Commutative Property
of Addition
2
1/
•
1
=
5
2
1/
5
2
1/
•
1
=
5
2
1/
5
Identity Property of
Multiplication
-8
2/
+
0
=
-8
5
2/
5
-8
2/
+
0
=
-8
5
2/
5
Identity Property of
Addition
2
[(- /
)(5)]9
=
3
2
-/
[(5)(9)]
3
2
[(- /
)(5)]9
=
3
2
-/
[(5)(9)]
3
Associative Property of
Multiplication
6(3 – 2n) = 18 – 12n
6(3 – 2n) = 18 – 12n
Distributive Property
2x + 3 = 3 + 2x
2x + 3 = 3 + 2x
Commutative Property
of Addition
ab = ba
ab = ba
Commutative Property
of Multiplication
a+0=a
a+0=a
Identity Property of
Addition
a(bc) = (ab)c
a(bc) = (ab)c
Associative Property of
Multiplication
a•1 = a
a•1 = a
Identity Property of
Multiplication
a +b = b + a
a +b = b + a
Commutative Property
of Addition
a(b + c) = ab + ac
a(b + c) = ab + ac
Distributive Property
a + (b + c) = (a +b) + c
a + (b + c) = (a +b) + c
Associative Property of
Addition
a + (-a) = 0
a + (-a) = 0
Inverse Property of
Addition
Properties of Real Numbers
Commutative
Associative
Distributive
Identity + ×
Inverse + ×