Properties of Real Numbers July 21, 2015

Objectives Identify the different properties of real numbers Write the missing number in an equation where a principal has been applied Simplify an expression by applying the property specified

Closure Property of Addition The sum of two real number is a real number. Example: 3 + 5 = 8 3 is real number, 5 is real number, the sum of 3 and 5 is real number

Closure Property of Multiplication The product of two real number is a real number Example: 3 x 5 = 15 3 is real number. 5 si real number. The product of 3 and 5 is real number.

Commutative Property of Addition = 7 + 5 5 + 7 = 3 + 8 8 + 3 4 + 6 = 6 + 4 21 + = 13 13 + 21 If we change the order of the addends, will the sum change? Changing the order of the addends does not affect the sum. a + b = b + a

Commutative Property of Multiplication = 2 x 3 3 x 2 = 5 x 4 4 x 5 7 x 6 = x 7 6 31 x = 11 11 x 31 If we change the order of the factors, does the product change? Changing the order of the factors does not affect the product. a x b = b x a

Associative Property of Addition ? 3 + 5 + 4 ( ) ( ) 3 + 5 + 4 = ? 3 + 9 = 8 + 4 . = 12 12 Does the sum change as the grouping of the addends changes? ___ No

Associative Property of Addition ? 9 + 7 + 5 ( ) ( ) 9 + 7 + 5 = ? 9 + 12 = 16 + 5 . 21 = 21 Changing the grouping of the addends does not change the sum a + (b + c) = (a + b) + c

Associative Property of Addition ? 7 6 + (7 + 3) (6 + __) + 3 = ? 6 10 = 13 3 __ + __ __ + __ . 16 = 16 __ __ Changing the grouping of the addends does not change the sum a + (b + c) = (a + b) + c

Associative Property of Addition ? 2 9 3 __ + (9 + 3) (2 + __) + __ = ? 2 12 = 11 3 __ + __ __ + __ . 14 = 14 __ __ Changing the grouping of the addends does not change the sum a + (b + c) = (a + b) + c

Associative Property of Multiplication ? 2 x 9 x 5 ( ) ( ) 2 x 9 x 5 = ? 2 x 45 = x 5 18 . = 90 90 Does the product change as the grouping of the factors changes? ___ No

Associative Property of Multiplication ? 4 x 5 x 3 ( ) ( ) 4 x 5 x 3 = ? 4 x 15 = 20 x 3 . 60 = 60 Changing the grouping of the factors does not change the product a x (b x c) = (a x b) x c

Associative Property of Multiplication ? 7 2 x (7 x 3) (2 x __) x 3 = ? 2 21 = 14 3 __ x __ __ x __ . 42 = 42 __ __ Changing the grouping of the factors does not change the product a x (b x c) = (a x b) x c

Associative Property of Multiplication ? 8 5 6 __ x (5 x 6) (8 x __) x __ = ? 8 30 = 40 6 __ x __ __ x __ . 240 = 240 __ __ Changing the grouping of the factors does not change the product a x (b x c) = (a x b) x c

Distributive Property of Multiplication with Respect to Addition 2 x (8 + 5) 2 x (8 + 5) (2 x 8) + (2 x 5) 2 x 16 + 10 13 26 26 What number was multiplied by 8 and 5? What operations are involved in the equation? Did you get the same answer?

Distributive Property of Multiplication with Respect to Addition 7 x (3 + 4) 7 x (3 + 4) (7 x 3) + (7 x 4) 21 + 28 7 x 7 49 49 We can break one of the factors as the sum of addends, multiply each of these addends by the factor then add their products a x (b + c) = (a x b) + (a x c)

Distributive Property of Multiplication with Respect to Addition 7 x (9 + 3) 7 9 7 3 7 x (9 + 3) (_ x _) + (_ x _) 7 12 63 21 __ x __ __ + __ 84 84 __ __ We can break one of the factors as the sum of addends, multiply each of these addends by the factor then add their products a x (b + c) = (a x b) + (a x c)

Identity Property of Addition = 7 + 0 7 = 0 + 8 8 4 + 0 = 4 21 + = 21 If we change the order of the addends, will the sum change? The sum of zero and a number or vice versa is equal to the number itself a + 0 = a zero The identity element for addition is ____

Identity Property of Multiplication = 7 x 1 7 = 1 x 43 43 79 x 1 = 79 83 x = 1 83 If we change the order of the addends, will the sum change? The product of one and a number or vice versa is equal to the number itself a x 1 = a one The identity element for multiplication is ____

Inverse Property of Addition 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.

Inverse Property of Multiplication For each real number a there exists a unique real number such that their product is 1.

Let’s play “Name that property!”

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

Identity Property of Addition 3. Which Property? 8 + 0 = 8 Identity Property of Addition

Commutative Property of Multiplication 5. Which Property? 6 • 4 = 4 • 6 Commutative Property of Multiplication

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

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

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

Closure Property of Addition 8. Which Property? even + even = even Closure Property of Addition

9. Which Property? 3(2 + 5) = 3•2 + 3•5 Distributive Law

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

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

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

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

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

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

2) 9 + (12 + 4) = ___________ APA 3) 35 x 1 = __________ IPM Complete each equation using the indicated property. 1) 4 + 21 = _________ CPA 2) 9 + (12 + 4) = ___________ APA 3) 35 x 1 = __________ IPM 4) 3 x (8 + 7) = ____________ DL 5) 32 x 56 = ________ CPM 21 + 4 (9 + 12) + 4 35 (3 x 8) + (3 x 7) 56 x 32