College Algebra 9th txtbk

SECTION R–1 Algebra and Real Numbers 7
MATCHED PROBLEM 2
Which real number property justifies the indicated statement?
(A) 4 (2 x) (4 2) x
(C) 3x 7x (3 7)x
(E) If ab 1, then b 1a.
(B)
(D)
(a b) c c (a b)
(2x 3y) 0 2x 3y
Z Further Operations and Properties
Subtraction of real numbers can be defined in terms of addition and the additive inverse. If
a and b are real numbers, then a b is defined to be a (b). Similarly, division can be
defined in terms of multiplication and the multiplicative inverse. If a and b are real numbers
and b 0, then a b (also denoted ab) is defined to be a b 1 .
Z DEFINITION 2 Subtraction and Division of Real Numbers
For all real numbers a and b:
Subtraction:
Division:
a b a (b)
a b a b 1 b 0
5 3 5 (3) 2
3 2 3 2 1 3 1 2 1.5
It is important to remember that
Division by 0 is never allowed.
ZZZ EXPLORE-DISCUSS 1
(A) Give an example that shows that subtraction of real numbers is not commutative.
(B) Give an example that shows that division of real numbers is not commutative.
The basic properties of the set of real numbers, together with the definitions of subtraction
and division, lead to the following properties of negatives and zero.
Z THEOREM 1 Properties of Negatives
For all real numbers a and b:
1. (a) a
2. (a)b (ab) a(b) ab
3. (a)(b) ab
4. (1)a a
a
5.
b 0
b a b a b
a
6. b 0
b a b
a b a b