Properties of Real Numbers

Properties of Real NumbersSOL 7.16VocabularyAddend: a number that is added to another Factor: a number that is being multiplied Sum: The answer to an addition problemProduct: The answer to a multiplication problem Difference: The answer to a subtraction problem Reciprocal: the multiplicative inverse of a number Make sure to learn these important termswe will use them from now on! Identity Element: numbers that combine with other numbers without changing the other numbersInverse: numbers that combine with other numbers and result in identity elementsCommutative Property ofAdditionMultiplicationChanging the order of the addends does not change the sum.Changing the order of the factors does not change the product.For Example:For Example:Lets think about the Commutative Property!What Changed from one side of the equal sign to the other? What should we look for to help us identify this property?The order of the addends or factors changed.

I think of commuters when I think of the commutative property. You start one way and end, but to start again you must leave where you are and go back to where you started. Complete the examples using the commutative property.a) 4 + 5 =b) 8 (3 - 2) =c) 6- ( 5+ 9) = d) fg=5 + 4(3 - 2)86-(9 + 5)gfCan you make your own example?Write down what you think of! Associative Property ofAdditionMultiplicationRegrouping the addends does not change the sum.Regrouping the factors does not change the product.For Example:For Example:Lets think about the Associative Property!What Changed from one side of the equal sign to the other? How is the associative property different from the commutative?The number inside of the grouping symbols or parentheses changed.

In the commutative property the order of the numbers changed, but the numbers that were in the parentheses changed with the associative. Complete the examples using the associative property.a) 2+(4 + 5) =b) 8(3 2) =c) 6+( 5+ 9) = d) a(bc) =(2 + 4) + 5(8 3) 2(6+5) + 9 (ab)cMake your own example!How will you remember this property?What aboutSubtraction?Division?Is subtraction commutative? Associative? Is division commutative? Associative?For Example:For Example:The Distributive PropertyThe product of a number and the sum (or difference) of two other numbers equals the sum (or difference) of theproducts of the number and each other number.For Example:Lets think about the Distributive Property!What Changed from one side of the equal sign to the other? What should we look for to help us identify this property?The number on the outside of the parentheses was passed out to the numbers inside the parentheses.

I think of distributing flyers to the class. You start with one stack of flyers, then once you distribute the stack to the class each person has that same flyer. Complete the examples using the distributive property.a) 2(4 + 5) =b) 8 (3 - 2) =c) 6( 5+ 9) = d) e(f-g)=(24) + (25) (8 3)- (82)(65) + (69)(ef) (eg)Write down what you think of! Can you make your own example?Additive IdentityInverseThe sum of any real number and zero is equal to the given real number.The sum of a number and its additive inverse (opposite) always equals zero.For Example:For Example:MultiplicativeIdentityInverseThe product of any real number and one is equal to that given number.The product of a number and its multiplicative inverse (reciprocal) always equals one.For Example:For Example:Lets think about the Identity Properties!What are the identity elements? How can we tell the identity properties from the inverse properties?The additive identity element is 0 and the multiplicative identity element is one.

The identity properties result in the same real number that was in the problem. Also, in the identity properties 0 and 1 are being added or multiplied. They are the answers in the inverse properties!Complete the examples using the identity properties.a) 4 + 0 =b) 8 (1) =c) 0+( 5+ 9) =

48(5+9)Make your own example of each.Lets think about the Inverse Properties!Is there a number with no additive inverse? What is the difference between the additive inverse and the multiplicative inverse?Zero does not have an opposite. It is neither positive or negative to have an additive inverse.With the additive inverse you are adding the opposite, but with multiplicative inverse you are multiplying the reciprocal, not the opposite.Complete the examples using the inverse properties.-42Make your own example of each.The Multiplicative Property of Zero!The product of any real number and zero is zero!For Example:Lets think about the Multiplicative Property of Zero!Both the Multiplicative Property of Zero and the Additive Identity both have an answer of zero. How will you remember which property is which? The multiplicative property of zero has a product of zero and there is a zero both on the right and the left of the equal sign.

The additive identity has a sum of zero and there is only a zero on one side of the equal sign.Complete the examples using the multiplicative property of zero.a) 2(0) =b) 0 = 0 c)( 5- 9) ___ = 0

0any real number0Can you make your own example?Properties ExamplesLets see if you can identify these properties!Commutative Property of MultiplicationMultiplicative InverseCommutative Property of AdditionAssociative Property of MultiplicationProperty of ZeroMultiplicative IdentityAssociative Property of AdditionDistributive PropertyAdditive InverseDistributive PropertyCommutative Property of MultiplicationMultiplicative InverseAdditive IdentityAssociative Property of AdditionDistributive Property