Properties of Numbers Objective: Recognize the properties of equality and identity. Recognize the Commutative and Associative Properties.

Properties of Equality and Identity Equivalent expressions are expressions that represent the same number. The properties below allow you to write an equivalent expression for a given expression.

Properties of Equality and Identity Properties of EqualityWordsSymbolsExamples Reflexive Property Symmetric Property Transitive Property Substitution Property Any quantity is equal to itself. For any number a, a = a. 5 = = If one quantity equals a second quantity, then the second quantity equals the first. For any numbers a and b, if a = b, then b = a. If 8 = 2 + 6, then = 8. If one quantity equals a second quantity and the second quantity equals a third quantity, then the first quantity equals the third quantity. For any numbers a, b, and c, if a = b and b = c, then a = c. If = and = 15, then = 15. A quantity may be substituted for its equal in any expression. If a = b, then a may be replaced by b in any expression If n = 11, then 4n = 4 11.

Properties of Equality and Identity The sum of any number and 0 is equal to the number. Thus, 0 is called the additive identity.

Properties of Equality and Identity Addition PropertiesWordsSymbolsExamples Additive Identity Additive Inverse For any number a, the sum of a and 0 is a. a + 0 = 0 + a = a = = 2 A number and its opposite are additive inverses of each other a + (-a) = (-3) = 0 4 – 4 = 0

Properties of Equality and Identity There are also special properties associated with multiplication. Consider the following equations. 4 n = 4 The solution of the equation is 1. Since the product of any number and 1 is equal to the number, 1 is called the multiplicative identity. 6 m = 0 The solution of the equation is 0. The product of any number and 0 is equal to 0. This is called the Multiplicative Property of Zero.

Properties of Equality and Identity Two numbers whose product is 1 are called multiplicative inverses or reciprocals. Zero has no reciprocal because any number times 0 is 0.

Properties of Equality and Identity Multiplication Properties WordsSymbolsExamples Multiplicative Identity Multiplication Property of Zero Multiplicative Inverse For any number a, the product of a and 1 is a. a 1 = a 1 a = a 14 1 = = 14 For any number a, the product of a and 0 is 0. a 0 = 0 0 a = = = 0 For every number a / b, where a, b ≠ 0, there is exactly one number b / a such that the product of a / b and b / a is 1.

Example 1 Evaluate ¼ (12 – 8) + 3(15 ÷ 5 – 2). Name the property used in each step. =¼ (4) + 3(15 ÷ 5 – 2) =¼ (4) + 3(3 – 2) =¼(4) + 3(1) =1 + 3(1) =1 + 3 =4=4 Substitution: 12 – 8 = 4 Substitution: 15 ÷ 5 = 3 Substitution: 3 – 2 = 1 Multiplicative Inverse: ¼ (4) = 1 Multiplicative Identity: 3(1) = 3 Substitution: = 4

Check Your Progress Choose the best answer for the following. Evaluate 1 / 3 (10 – 7) + 4(18 ÷ 9 – 1) A.4 B.5 C.1 D.0 = 1 / 3 (3) + 4(18 ÷ 9 – 1) = 1 / 3 (3) + 4(2 – 1) = 1 / 3 (3) + 4(1) = 1 + 4(1) = 1 + 4

Commutative and Associative Properties PropertyWordsSymbolsExamples Commutative Property Associative Property The order in which you add or multiply numbers does not change their sum or product. For any numbers a and b, a + b = b + a and a b = b a = = 11 7 The way you group three or more numbers when adding or multiplying does not change their sum or product. For any numbers a, b, and c, (a + b) + c = a + (b + c) and (ab)c = a(bc). (3 + 5) + 7 = 3 + (5 + 7) (2 6) 9 = 2 (6 9)

Example 2 Migina made a list of trail lengths to find the total miles she rode. Find the total miles Migina rode her horse miles ( ) Trails NameMiles Bent Tree4.25 Knob Hill6.50 Meadowrun9.00 Pinehurst7.75

Check Your Progress Choose the best answer for the following. Darlene rode the city train from the Winchester Street Station to the airport. How far did she travel on the train? A.4.5 mi B.5.5 mi C.6.0 mi D.6.2 mi Station Distance from Previous Station (mi) Winchester St.0.0 Reynolds Sq.0.6 Idris St.1.2 Pinehurst1.8 Airport ( ) =

Example 3 Evaluate using the properties of numbers. Name the property used in each step. = =(2 5) (7 8) =10 56 =560 Commutative Property (mult) Associative Property (mult) Substitution

Check Your Progress Choose the best answer for the following. Evaluate A.45 B.36 C.15 D.180 = = (3 3) (4 5) = 9 20