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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. ab c ab ac 32 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 + ×