Properties of Real NumbersProperties of Real Numbers

CommutativeCommutative

AssociativeAssociative

DistributiveDistributive

Identity + Identity + ××

Inverse + Inverse + ××

Commutative PropertiesCommutative Properties

Changing the Changing the orderorder of the numbers in of the numbers in addition or multiplication will not addition or multiplication will not change the result.change the result. Commutative Property of AdditionCommutative Property of Addition states: 2 + 3 = 3 + 2 or a + b = b + a.states: 2 + 3 = 3 + 2 or a + b = b + a. Commutative Property of Commutative Property of MultiplicationMultiplication states: 4 states: 4 • 5 = 5 • 4 or • 5 = 5 • 4 or abab = = ba.ba.

Associative PropertiesAssociative Properties

Changing the Changing the groupinggrouping of the of the numbers in addition or multiplication numbers in addition or multiplication will not change the result.will not change the result.

Associative Property of AdditionAssociative Property of Addition states: 3 + (4 + 5)= (3 + 4)+ 5 or states: 3 + (4 + 5)= (3 + 4)+ 5 or aa + ( + (bb + + cc)= ()= (aa + + bb)+ )+ cc

Associative Property of MultiplicationAssociative Property of Multiplication states: (2 states: (2 • 3) • 4 = 2 • (3 • 4) or • 3) • 4 = 2 • (3 • 4) or (ab)c = a(bc)(ab)c = a(bc)

Distributive Property Distributive Property

Multiplication Multiplication distributesdistributes over addition over addition..

acabcba

5323523

Additive Identity PropertyAdditive Identity Property

There exists a unique number 0 such There exists a unique number 0 such that zero preserves identities under that zero preserves identities under addition.addition.

aa + 0 = + 0 = aa and 0 + and 0 + a = aa = aIn other words adding zero to a In other words adding zero to a number does not change its value.number does not change its value.

Multiplicative Identity PropertyMultiplicative Identity Property

There exists a unique number 1 such There exists a unique number 1 such that the number 1 preserves identities that the number 1 preserves identities under multiplication.under multiplication.

aa ∙ 1∙ 1 = = aa and 1 and 1 ∙∙ a = aa = aIn other words multiplying a number In other words multiplying a number by 1 does not change the value of the by 1 does not change the value of the number.number.

Additive Inverse PropertyAdditive Inverse Property

For each real number For each real number aa there there exists a unique real number –exists a unique real number –aa such that their sum is zero. such that their sum is zero.

aa + (- + (-aa) = 0) = 0In other words opposites add to In other words opposites add to zero.zero.

Multiplicative Inverse PropertyMultiplicative Inverse Property

For each real number For each real number aa there exists a there exists a

unique real number such that their unique real number such that their

product is 1. product is 1.

1

a

11

a

a

Let’s play “Name that property!”

State the property or properties that State the property or properties that justify the following.justify the following.

3 + 2 = 2 + 33 + 2 = 2 + 3

Commutative Property

State the property or properties that State the property or properties that justify the following.justify the following.

10(1/10) = 110(1/10) = 1

Multiplicative Inverse Property

State the property or properties that State the property or properties that justify the following.justify the following.

3(x – 10) = 3x – 30 3(x – 10) = 3x – 30

Distributive Property

State the property or properties that State the property or properties that justify the following.justify the following.

3 + (4 + 5) = (3 + 4) + 53 + (4 + 5) = (3 + 4) + 5 Associative Property

State the property or properties that State the property or properties that justify the following.justify the following.

(5 + 2) + 9 = (2 + 5) + 9(5 + 2) + 9 = (2 + 5) + 9

Commutative Property

3 + 7 = 7 + 3

Commutative Commutative Property of AdditionProperty of Addition

2.2.

8 + 0 = 8

Identity Property of Identity Property of AdditionAddition

3.3.

6 • 4 = 4 • 6

Commutative Property Commutative Property of Multiplicationof Multiplication

5.5.

5 • 1 = 5

Identity Property of Identity Property of MultiplicationMultiplication

11.11.

51/7 + 0 = 51/7

Identity Property of Identity Property of AdditionAddition

25.25.

a + (-a) = 0

Inverse Property of Inverse Property of AdditionAddition

40.40.

Properties of Real NumbersProperties of Real Numbers

CommutativeCommutative

AssociativeAssociative

DistributiveDistributive

Identity + Identity + ××

Inverse + Inverse + ××