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Properties of Real Numbers

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List of Properties of Real Numbers Commutative Associative Distributive Identity Inverse

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

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

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

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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)

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Distributive Property Multiplication distributes over addition.

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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

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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

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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

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Multiplicative Inverse Property

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Let’s play “Name that property!”

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State the property or properties that justify the following. 3 + 2 = 2 + 3 Commutative Property

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State the property or properties that justify the following. 10(1/10) = 1 Multiplicative Inverse Property

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State the property or properties that justify the following. 3(x – 10) = 3x – 30 Distributive Property

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State the property or properties that justify the following. 3 + (4 + 5) = (3 + 4) + 5 Associative Property

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State the property or properties that justify the following. (5 + 2) + 9 = (2 + 5) + 9 Commutative Property

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3 + 7 = 7 + 3 Commutative Property of Addition 2.

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8 + 0 = 8 Identity Property of Addition 3.

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6 4 = 4 6 Commutative Property of Multiplication 5.

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

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

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

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even + even = even Closure Property 8.

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3(2 + 5) = 32 + 35 Distributive Property 9.

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

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5 1 = 5 Identity Property of Multiplication 11.

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

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

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

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

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

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-8 + 0 = -8 Identity Property of Addition 18.

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

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

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

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(5 + 4)9 = 45 + 36 Distributive Property 22.

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

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

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5 1 / 7 + 0 = 5 1 / 7 Identity Property of Addition 25.

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

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

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

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-8 2 / 5 + 0 = -8 2 / 5 Identity Property of Addition 29.

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

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

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

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ab = ba Commutative Property of Multiplication 33.

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a + 0 = a Identity Property of Addition 34.

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

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a1 = a Identity Property of Multiplication 36.

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

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

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

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

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

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