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Algebraic Properties Learning Goal: The student will be able to summarize properties and make connections between real number operations.

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The Commutative Property The ORDER in which you add or multiply two numbers does not affect the result. Commutative Property of Addition a + b = b + a 3 + 7 = 7 + 3 Commutative Property of Multiplication ab = ba 7●3 = 3●7

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The Associative Property When you add or multiply any three real numbers, the GROUPING (or association) of the numbers does not affect the result. Associative Property of Addition a + (b + c) = (a + b) + c 5 + (6 + 7) = (5 + 6) + 7 Associative Property of Multiplication a(bc) = (ab)c 2(3●4)= (2●3)4 Note: when values “leave” the parentheses but do NOT affect the value

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The Additive Identity Property 1.If 0 is added to any real number, the sum is equal to the given number. 2. When you multiply any given real number by 1, the product is equal to the given number. Note: Think of a mirror image. Identity Property of Addition a + 0 = a 6 + 0 = 6 Identity Property of Multiplication a ● 1 = a 2 ● 1= 2

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Distributive Property When you distribute something, you separate it into portions. The distributive property states that for any numbers a, b, and c: a(b + c) = (ab) + (ac) a(b – c) = (ab) – (ac) Example: -4(5 + 3) = (-4 x 5) + (-4 x 3) -20 + -12 = -32 -(3 – 2) = (-1 x 3) –(-1 x 2) -3 – (-2) = -1

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The Inverse Property The sum of a number and its opposite is zero. The product of a number and it’s reciprocal is 1. Inverse Property of Addition a + -a = 0 2 + -2 = 0 Inverse Property of Multiplication a ● 1/a = 1 2 ● ½ = 1

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The Zero Property When you multiply a real number by 0, the resulting product is zero. Zero Property a ● 0 = 0

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Try this…. 1. x + 9 = 9 + x is an example of which property? 4. -22 + 22 = 0 is an example of which property? 2. x + ( y + 3 ) = x + ( 3 + y ) is an example of which property? 3. ( 5 y ) ( 1 ) = 5 y is an example of which property? Commutative Property for Addition Identity Property for Multiplication Inverse Property for Addition

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Try this Cont.…. 5. 5 1 = 5 is an example of which property? 8. is an example of which property? 6. ( x y ) z = x ( y z ) is an example of which property? 7. 10 0 = 0 is an example of which property? Identity Property for Multiplication Associative Property for Multiplication Zero Property for Multiplication Inverse Property for Multiplication

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The Closure Property Closure Property of Addition the sum of any two real numbers is also a real number and there is a unique (only one) solution Example: The sum of any two even numbers is always even Example: If you multiply a whole number by 2, your product is always even Closure Property of Multiplication the product of any two real numbers is a real number and there is a unique (only one) solution

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More about Closure… Let's check out this question. If you divide two even numbers (from the set of even numbers), is the quotient even? When you find even ONE example that does not work, the set is not closed under that operation. The even numbers are not closed under division. 12 / 6 = 2 Yes, 2 is even. 24 / 2 = 12 Yes, 12 is even. 100 / 4 = 25 NO, 25 is not even!

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