Properties of Real Numbers

List of Properties of Real Numbers Commutative Associative Distributive Identity Inverse

Commutative Properties Definition: Changing the order of the numbers in addition or multiplication will not change the result.

Commutative Properties PropertyNumbersAlgebra Addition2 + 3 = 3 + 2a + b = b + a Multiplication4 5 = 5 4ab = ba

Associative Properties Definition: Changing the grouping of the numbers in addition or multiplication will not change the result.

Associative Properties PropertyNumbersAlgebra Addition 3 + (4 + 5)= (3 + 4)+ 5 a + (b + c)= (a + b)+ c Multiplication (2 3) 4 = 2 (3 4) (ab)c = a(bc)

Distributive Property Multiplication distributes over addition.

Additive Identity Property Definition: Zero preserves identities under addition. In other words, adding zero to a number does not change its value. Example: a + 0 = a and 0 + a = a

Multiplicative Identity Property Definition: The number 1 preserves identities under multiplication. In other words, multiplying a number by 1 does not change the value of the number. Example: a ∙ 1 = a and 1 ∙ a = a

Additive Inverse Property Definition: For each real number a there exists a unique real number –a such that their sum is zero. In other words, opposites add to zero. Example: a + (-a) = 0

Multiplicative Inverse Property

Let’s play “Name that property!”

State the property or properties that justify the following = Commutative Property

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 Distributive Property

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 Commutative Property

3 + 7 = Commutative Property of Addition 2.

8 + 0 = 8 Identity Property of Addition 3.

6 4 = 4 6 Commutative Property of Multiplication 5.

17 + (-17) = 0 Inverse Property of Addition 6.

2(5) = 5(2) Commutative Property of Multiplication 7.

(2 + 1) + 4 = 2 + (1 + 4) Associative Property of Addition 1.

even + even = even Closure Property 8.

3(2 + 5) = Distributive Property 9.

6(78) = (67)8 Associative Property of Multiplication 10.

5 1 = 5 Identity Property of Multiplication 11.

(6 – 3)4 = 64 – 34 Distributive Property 13.

1(-9) = -9 Identity Property of Multiplication 14.

3 + (-3) = 0 Inverse Property of Addition 15.

1 + [-9 + 3] = [1 + (-9)] + 3 Associative Property of Addition 16.

-3(6) = 6(-3) Commutative Property of Multiplication 17.

= -8 Identity Property of Addition 18.

37 – 34 = 3(7 – 4) Distributive Property 19.

6 + [(3 + (-2)] = (6 + 3) + (- 2) Associative Property of Addition 20.

7 + (-5) = Commutative Property of Addition 21.

(5 + 4)9 = Distributive Property 22.

-3(5 4) = (-3 5)4 Associative Property of Multiplication 23.

-8(4) = 4(-8) Commutative Property of Multiplication 24.

5 1 / = 5 1 / 7 Identity Property of Addition 25.

3 / 4 – 6 / 7 = – 6 / / 4 Commutative Property of Addition 26.

1 2 / 5 1 = 1 2 / 5 Identity Property of Multiplication 27.

(fraction)(fraction) = fraction Closure Property 28.

-8 2 / = -8 2 / 5 Identity Property of Addition 29.

[(- 2 / 3 )(5)]9 = - 2 / 3 [(5)(9)] Associative Property of Multiplication 30.

6(3 – 2n) = 18 – 12n Distributive Property 31.

2x + 3 = 3 + 2x Commutative Property of Addition 32.

ab = ba Commutative Property of Multiplication 33.

a + 0 = a Identity Property of Addition 34.

a(bc) = (ab)c Associative Property of Multiplication 35.

a1 = a Identity Property of Multiplication 36.

a +b = b + a Commutative Property of Addition 37.

a(b + c) = ab + ac Distributive Property 38.

a + (b + c) = (a +b) + c Associative Property of Addition 39.

a + (-a) = 0 Inverse Property of Addition 40.

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