1. MTH 11203 Algebra PROPERTIES OF THE REAL NUMBER SYSTEM CHAPTER 1 SECTION 10

2. If a and b represent any real numbers, then a + b = b + a Commutative property involves a change in order. The order that you add does not matter, same results Exp: 5 + 2 = 2+ 5 7 = 7 Exp:-3 + (-5) = -5 + (-3) -8 = -8 Commutative property will not work for subtraction Commutative Property of Addition

3. If a and b represent any real numbers, then a · b = b · a The order that you multiply does not matter, same results Exp: (5)(6) = (6)(5) 30 = 30 Exp:(r)(g) = (g)(r) rg = rg Commutative property will not work for division Commutative property changes the order Commutative Property of Multiplication

4. If a and b represent any real numbers, then (a + b) + c = a + (b + c) The associative property involves a change in grouping. The order that you add does not matter, same results Exp: (2 + 3) + 4 = 2 + (3 + 4) Exp: 6 + (w + 1) = (6 + w) + 1 5 + 4 = 2 + 7 7 + w = 7 + w 9 = 9 Exp:(3 + 4) + x = 3 + (4 + x) 7 + x = 7 + x Associative property will not work for subtraction Associative Property of Addition

5. If a and b represent any real numbers, then (a · b) · c = a · (b · c) The order that you multiply does not matter, same results Exp: (2 · 6) · 4 = 2 · (6 · 4) 12 · 4 = 2 · 24 48 = 48 Exp:(3 · 7) · x = 3 · (7 · x) 21 · x = 21 · x 21x = 21x Associative property will not work for division The associative property changes grouping Associative Property of Multiplication

6. If a and b represent any real numbers, then a(b + c) = ab + ac The order that you multiply does not matter, same results Exp: 3(2 + 4) = (3)(2) + (3)(4) (3)(6) = 6 + 12 18 = 18 Exp:6(x – 9) = (6)(x) – (6)(9) 6x – (6)(9) = 6x – 54 6x – 54 = 6x – 54 Expand: a (b + c + d + e + f + … + n) = ab + ac + ad + ae + af + … + an Distributive Property

7. Identity Property of Addition If a and b represent any real numbers, then a + 0 = a and 0 + a = a Zero is the identity element of addition Exps: 3 + 0 = 3 6 + 0 = 6 50 + 0 = 50 Identity Property of Multiplication If a and b represent any real numbers, then a · 1 = a and 1 · a = a One is the identity element of Multiplication Exp:3 · 1 = 3 6 · 1 = 6 50 · 1 = 50 Identity Property

8. Inverse Property of Addition If a and b represent any real numbers, then a + (-a) = 0 and -a + a = 0 Additive inverses are any two numbers whose sum is 0 or opposites. Exp: 3 + (-3) = 0 6 + (-6) = 0 50 + (-50) = 0 Inverse Property of Multiplication If a and b represent any real numbers, then a · = 1 and · a = 1 a ≠ 0 Multiplicative inverses are any two numbers whose product is 1 or reciprocal. Exp:3 · = 1 6 · = 1 50 · = 1 Inverse Property

9. #29 pg 84) p · (q · r) = (p · q) · r Associative Property of Multiplication #24 pg 84)3 + y = y + 3 Commutative Property of Addition #31 pg 84)4(d + 3) = 4d + 12 Distributive Property #28 pg 84)-4x + 4x = 0 Inverse Property of Addition #33 pg 85) 3z · 1 = 3z Identity Property of Multiplication Name the property

10. Exp:4(x – 7) = 4x + (4)(-7) 4(x – 7) = 4x – 28 Distributive Property Exp:5y + (-5y) = 0 Inverse Property of Addition Name the property

11. Exp: (3 · 6) · 9 = (3 · 6) · 9 = 3 · (6 · 9) Associative Property of Multiplication Exp:7(x – 4) = 7(x – 4) = 7x – 28 Distributive Property #33 pg 85) 6b + 0 = 6b + 0 = 6b Identity Property of Addition Complete the equation and name the property

12. HOMEWORK 1.10 Page 84 – 85 #23, 25, 27, 37, 39, 41, 43, 53