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Notes: Properties of Numbers 1.3: I can apply algebraic properties to simplify algebraic expressions. (C.1.c) __________________________________________________________ Equivalent Expressions: ____________________________________________________ The properties below allow you to write an equivalent expression for a given expression. Properties of Equality Symbols For any number a, a = a. Property Reflexive Property Words Any quantity is equal to itself. Symmetric Property 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. Transitive Property 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. A quantity may be substituted for its equal in any expression. If a = b, then a may be replaced by b in any expression. Substitution Property Examples Key Words Name the Property of Equality illustrated in each example below. 1. The distance from Sarah’s house to the school is equal to the distance from the school to Sarah’s house. 1 2. If the area of a triangle is calculated by 2 𝑏ℎ. If you have a triangle with a base of 5 ft. and a height of 10 1 ft. you can calculate the area by solving (5)(10). 2 3. Having 2 apples plus 8 apples is the same as having 10 apples, and having 10 apples is the same as having 2 apples plus 8 apples. 4. If two quarters is equal to five dimes, and five dimes is equal to ten nickels, then two quarters is equal to ten nickels. There are special properties associated with both addition and multiplication. Property Words Commutative The order in which Property you add or multiply numbers does not change the sum or product. Symbols For any numbers a and b, a+b = b+a and a•b = b•a Associative Property The way you group three or more numbers when adding or multiplying does not change their sum or product. Property Additive Identity Or Identity Property of Addition Multiplicative Identity Or Identity Property of Multiplication Words For any number a, the sum of a and 0 is a. Property Additive Inverse Or Inverse Property of Addition Words A number and its opposite are additive inverses of each other. Multiplicative Inverse Or Inverse Property of Multiplication For every number , 𝑏 where a ≠ 0 and b ≠ 0, there is exactly 𝑏 one number 𝑎 such For any number a, the product of a and 1 is a. 𝑎 𝑎 that the product of 𝑏 𝑏 and 𝑎 is 1. Examples Key Words For and numbers a, b, and c, (a+b)+c = a+(b+c) and (a•b)•c = a•(b•c) - Identity Properties Symbols a+0=a 0+a=a Examples Key Words Examples Key Words a•1=a 1•a=a - Inverse Properties Symbols a + (-a) = 0 a–a=0 𝑎 𝑏 •𝑎=1 𝑏 𝑏 𝑎 •𝑏=1 𝑎 Property Words Multiplication For any number a, Property of the product of a and Zero 0 is 0. Name the Property illustrated in each example below. 5. 5 + 0 = 5 6. 8 + 6 = 6 + 8 1 7. 5 ∙ 5 = 1 8. (9 ∙ 5) ∙ 2 = 9 ∙ (5 ∙ 2) 9. 12 ∙ 1 = 12 10. 4 + (−4) = 0 11. 7 ∙ 0 = 0 12. Property of Zero Symbols a•0=0 0•a=0 2 3 ∙ =1 3 2 13. 8 ∙ 2 = 2 ∙ 8 14. 0 + 14 = 14 15. 9 + (8 + 7) = (9 + 8) + 7 16. 3 − 3 = 0 Examples Key Words Evaluating Expressions using Properties of Numbers Properties Checklist for Evaluating Expressions Identity Property of Addition Identity Property of Multiplication Inverse Property of Addition Inverse Property of Multiplication Multiplication Property of Zero Substitution Property Evaluate each expression. Name the property used in each step. 1 17. 7(4 − 3) − 1 + 5 ∙ 5 1 7 18. 7 ∙ + 6(15 ÷ 3 − 5) 19. 7 + (9 − 32 ) 1 20. 6 ∙ 6 + 5(12 ÷ 4 − 3)