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.

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 such that their product is 1.

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

Which Property? 3 + 7 = 7 + 3

Commutative Property of Addition Which Property? 3 + 7 = 7 + 3 Commutative Property of Addition

Which Property? 8 + 0 = 8

Identity Property of Addition Which Property? 8 + 0 = 8 Identity Property of Addition

Which Property? 6 • 4 = 4 • 6

Commutative Property of Multiplication Which Property? 6 • 4 = 4 • 6 Commutative Property of Multiplication

Which Property? 17 + (-17) = 0

Inverse Property of Addition Which Property? 17 + (-17) = 0 Inverse Property of Addition

Which Property? 2(5) = 5(2)

Commutative Property of Multiplication Which Property? 2(5) = 5(2) Commutative Property of Multiplication

Which Property? (2 + 1) + 4 = 2 + (1 + 4)

Associative Property of Addition Which Property? (2 + 1) + 4 = 2 + (1 + 4) Associative Property of Addition

Which Property? even + even = even

Which Property? 3(2 + 5) = 3•2 + 3•5

Distributive Property Which Property? 3(2 + 5) = 3•2 + 3•5 Distributive Property

Which Property? 6(7•8) = (6•7)8

Associative Property of Multiplication Which Property? 6(7•8) = (6•7)8 Associative Property of Multiplication

Which Property? 5 • 1 = 5

Identity Property of Multiplication Which Property? 5 • 1 = 5 Identity Property of Multiplication

Properties Using Negatives

Which Property? (6 – 3)4 = 6•4 – 3•4

Distributive Property Which Property? (6 – 3)4 = 6•4 – 3•4 Distributive Property

Which Property? 1(-9) = -9

Identity Property of Multiplication Which Property? 1(-9) = -9 Identity Property of Multiplication

Which Property? 3 + (-3) = 0

Inverse Property of Addition Which Property? 3 + (-3) = 0 Inverse Property of Addition

Which Property? 1 + [-9 + 3] = [1 + (-9)] + 3

Associative Property of Addition Which Property? 1 + [-9 + 3] = [1 + (-9)] + 3 Associative Property of Addition

Which Property? -3(6) = 6(-3)

Commutative Property of Multiplication Which Property? -3(6) = 6(-3) Commutative Property of Multiplication

Which Property? -8 + 0 = -8

Identity Property of Addition Which Property? -8 + 0 = -8 Identity Property of Addition

Which Property? 3•7 – 3•4 = 3(7 – 4)

Distributive Property Which Property? 3•7 – 3•4 = 3(7 – 4) Distributive Property

Which Property? 6 + [(3 + (-2)] = (6 + 3) + (- 2)

Associative Property of Addition Which Property? 6 + [(3 + (-2)] = (6 + 3) + (- 2) Associative Property of Addition

Which Property? 7 + (-5) = -5 + 7

Commutative Property of Addition Which Property? 7 + (-5) = -5 + 7 Commutative Property of Addition

Which Property? (5 + 4)9 = 45 + 36

Distributive Property Which Property? (5 + 4)9 = 45 + 36 Distributive Property

Which Property? -3(5 • 4) = (-3 • 5)4

Associative Property of Multiplication Which Property? -3(5 • 4) = (-3 • 5)4 Associative Property of Multiplication

Which Property? -8(4) = 4(-8)

Commutative Property of Multiplication Which Property? -8(4) = 4(-8) Commutative Property of Multiplication

Properties Using Fractions

Which Property? 51/7 + 0 = 51/7

Identity Property of Addition Which Property? 51/7 + 0 = 51/7 Identity Property of Addition

Which Property? 3/4 – 6/7 = – 6/7 + 3/4

Commutative Property of Addition Which Property? 3/4 – 6/7 = – 6/7 + 3/4 Commutative Property of Addition

Which Property? 12/5 • 1 = 12/5

Identity Property of Multiplication Which Property? 12/5 • 1 = 12/5 Identity Property of Multiplication

Which Property? -8 2/5 + 0 = -8 2/5

Identity Property of Addition Which Property? -8 2/5 + 0 = -8 2/5 Identity Property of Addition

Which Property? [(-2/3)(5)]9 = -2/3[(5)(9)]

Associative Property of Multiplication Which Property? [(-2/3)(5)]9 = -2/3[(5)(9)] Associative Property of Multiplication

Properties Using Variables

Which Property? 6(3 – 2n) = 18 – 12n

Distributive Property Which Property? 6(3 – 2n) = 18 – 12n Distributive Property

Which Property? 2x + 3 = 3 + 2x

Commutative Property of Addition Which Property? 2x + 3 = 3 + 2x Commutative Property of Addition

Which Property? ab = ba

Commutative Property of Multiplication Which Property? ab = ba Commutative Property of Multiplication

Which Property? a + 0 = a

Identity Property of Addition Which Property? a + 0 = a Identity Property of Addition

Which Property? a(bc) = (ab)c

Associative Property of Multiplication Which Property? a(bc) = (ab)c Associative Property of Multiplication

Which Property? a•1 = a

Identity Property of Multiplication Which Property? a•1 = a Identity Property of Multiplication

Which Property? a +b = b + a

Commutative Property of Addition Which Property? a +b = b + a Commutative Property of Addition

Which Property? a(b + c) = ab + ac

Distributive Property Which Property? a(b + c) = ab + ac Distributive Property

Which Property? a + (b + c) = (a +b) + c

Associative Property of Addition Which Property? a + (b + c) = (a +b) + c Associative Property of Addition

Which Property? a + (-a) = 0

Inverse Property of Addition Which Property? a + (-a) = 0 Inverse Property of Addition

Properties of Real Numbers Commutative Associative Distributive Identity + × Inverse + ×