Principles Principles are basic truths or laws.

Luke 6:31 “And as ye would that men should do to you, do ye also to them likewise.”

Closure Property of Addition For all integers a and b, a + b is an integer.

Integers Integers are closed for addition. The set of odd numbers is not closed for addition.

Commutative Property of Addition For all integers a and b, a + b = b + a.

−5 + 3

Equivalent When two expressions represent the same value, they are said to be equivalent.

Example 1 Use the Commutative Property to write an equivalent expression for each of the following. Be sure the answer is in simplest form. a. −9 + 4 = 4 + (−9) = 4 − 9

Example 1 b. x + 9 = 9 + x

Associative Property of Addition For all integers a, b, and c a + (b + c) = (a + b) + c.

Example 2 Use the Associative Property to write an equivalent expression for each of the following. a. (−8 + 17) + 43 =

Example 2 b. −9 + (7 + x) =

Identity Property of Addition For any integer a, a + 0 = 0 + a = a.

Additive Inverse Additive inverses are two numbers whose sum is zero. A.K.A. “Opposites”

Example 3 What are the additive inverses of 17 and −14?

Example 4 Use the properties of addition to find the value of each variable. Name the properties used. a. x = 6 + (−6)

Example 4 b. y + 0 = 19 c. (9 + 2) − 6 = x + (2 − 6)

Property of Addition Description Example Closure: for all integers a and b, a + b is an integer. The sum of any two integers is an integer. 5 + 9 = 14; 14 is an integer. Commutative: a + b = b + a Changing the order of the addends does not change the sum. 8 + 4 = 4 + 8 Associative: (a + b) + c = a + (b + c) Changing the grouping of the addends does not change the sum. (−5 + 4) + 6 = −5 + (4 + 6)

Property of Addition Description Example Identity: a + 0 = 0 + a = a The sum of any integer and zero equals the original integer. −2 + 0 = 0 + (−2) = −2 Inverse: a + (−a) = −a + a = 0 The sum of any integer and its additive inverse equals zero, the identity element of addition. 6 + (−6) = −6 + 6 = 0

Exercise + a b c

Exercise Is this set closed? Why?

Exercise Is this set commutative? Why?

Exercise Which element is the identity element? Why?

Exercise What is the inverse of a? Why?

Exercise A clock has no zero on it. What number serves as the identity number for adding clock times?

Exercise Based on your answer to the last question, what is the inverse of 7 on a clock?