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Properties of Real Numbers The properties of real numbers help us simplify math expressions and help us better understand the concepts of algebra.

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Commutative Property of Addition a + b = b + a Example:7 + 3 = 3 + 7 Two real numbers can be added in either order to achieve the same sum.

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Commutative Property of Multiplication a x b = b x a Example:3 x 7 = 7 x 3 Two real numbers can be multiplied in either order to achieve the same product.

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Associative Property of Multiplication (a x b) x c = a x (b x c) Example: (6 x 4) x 5 = 6 x (4 x 5) When three real numbers are multiplied, it makes no difference which are multiplied first. Notice how multiplying the 4 and 5 first makes completing the problem easier.

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Associative Property of Addition (a + b) + c = a + (b + c) Example: (29 + 13) + 7 = 29 + (13 + 7) When three real numbers are added, it makes no difference which are added first. Notice how adding the 13 + 7 first makes completing the problem easier mentally.

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Additive Identity Property a + 0 = a Example: 9 + 0 = 9 The sum of zero and a real number equals the number itself. Memory note: When you add zero to a number, that number will always keep its identity.

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Multiplicative Identity Property a x 1 = a Example: 8 x 1 = 8 The product of one and a number equals the number itself. Memory note: When you multiply any number by one, that number will keep its identity.

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Additive Inverse Property a + (-a) = 0 Example: 3 + (-3) = 0 The sum of a real number and its opposite is zero.

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Multiplicative Inverse

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Property of Zero (Multiplication) When any number is multiplied with zero, the answer is zero. 98,756,432 X 0 = 0

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Property of Opposites a + (-a) = 0 If you added opposite #’s and ended with 0

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Distributive Property a(b + c) = ab + ac ora(b – c) = ab – ac Example: 2(3 + 4) = (2 x 3) + (2 x 4) or 2(3 - 4) = (2 x 3) - (2 x 4) Distributive Property is the sum or difference of two expanded products.

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Properties of Equality Addition property of equality If a = b, then a + c = b + c. Adding the same number to both sides of an equation does not change the equality of the equation. Subtraction property of equality If a = b, then a – c = b – c. Subtracting the same number from both sides of an equation does not change the equality of the equation. Multiplication property of equality If a = b and c ≠ 0, then a c = b c. Multiplying both sides of the equation by the same number, other than 0, does not change the equality of the equation. Division property of equality If a = b and c ≠ 0, then a ÷ c = b ÷ c. Dividing both sides of the equation by the same number, other than 0, does not change the equality of the equation.

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Transitive Property If a = b and b = c, then a = c If one quantity equals a second quantity and the second quantity equals a third quantity, then the first equals the third. If 1000 mm = 100 cm and 100 cm = 1 m, Then 1000 mm = 1m

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Symmetric Property If a + b = c then c = a + b If one quantity equals a second quantity, then the second quantity equals the first. If 10 = 4 + 6, then 4 + 6 = 10

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Reflexive Property a = a a + b = a + b Any quantity is equal to itself. 7 = 7 2 + 3 = 2 + 3

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Justify Steps in Solving Equations http://www.phschool.com/atschool/academy123/englis h/academy123_content/wl-book-demo/ph-374s.html http://www.phschool.com/atschool/academy123/englis h/academy123_content/wl-book-demo/ph-374s.html

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