1. Properties of Numbers A property is something that is true for all situations.

2. We will discuss four Properties 1.Commutative 2.Associative 3.Identity properties of one and zero 4.Inverse property

3. Commutative Property of addition and multiplication Order doesn’t matter a b = b a or a + b = b + a addition 5a + 4 = 4 + 5a multiplication 3 8 5b = 5b 3 8 In Algebraic Expressions

4. Commutative Property X + Y = Y + X Think of the elements as "commuting" from one location to another. "They get in their cars and drive to their new locations." This explanation will help you to remember that the elements are "moving" (physically changing places). =

5. Associative Property of multiplication and Addition Associative Property  (a · b) · c = a · (b · c) Example: (6 · 4) · 3 = 6 · (4 · 3) Associative Property  (a + b) + c = a + (b + c) Example: (6 + 4) + 3 = 6 + (4 + 3) In Algebraic Expressions addition (4x + 2x) + 7x = 4x + (2x + 7x) multiplication 2x 2 (3y) = 3y(2x 2 )

6. Associative Property (X + Y) + Z = X + (Y + Z) The associative property can be thought of as illustrating "friendships" (associations). The parentheses show the grouping of two friends. In the example below, the red girl (y) decides to change from the blue boyfriend (x) to the green boyfriend (z). "I don't want to associate with you any longer!" Notice that the elements do not physically move, they simply change the person with whom they are "holding hands" (illustrated by the parentheses.)

7. Identity Properties If you add 0 to any number, the number stays the same: a + 0 = a or 5 + 0 = 5 If you multiply any number times 1, the number stays the same: a x 1 = a or 5 x 1 = 5 In Algebraic Expressions Addition 5y + 0 = 5y Multiplication 2c × 1 = 2c

8. Identity Property X + 0 = X Additive Identity X 1 = X Multiplicative Identity Try to remember the " I " in the word identity. Variables can often times have an "attitude". " I am the most important thing in the world and I do not want to change!" The identity element allows the variable to maintain this attitude. Additive Identity is 0. Multiplicative Identity is 1.

9. Inverse Property Inverse Properties state that when a number is combined with its inverse, it is equal to its identity. There are two types of inverses of a number: Additive Inverse and Multiplicative Inverse. `- a` is said to be additive inverse of `a` if a + (- a) = 0. `1/a` is said to be multiplicative inverse of `a` if a 1/a = 1

10. Inverse Property What brings you back to the identity element using that operation? X + -X = 0 Additive Inverse X 1/X = 1 Multiplicative Inverse Think of the inverse as "inventing" an identity element. What would you need to add (multiply) to this element to turn it into an identity element? The Additive Inverse is the negation of the element. The Multiplicative Inverse is one divided by the element.