66 CHAPTER 1 The Real Number System
Here is a summary of the properties of real numbers discussed in this section.
Properties of Addition and MultiplicationFor any real numbers a, b, and c, the following properties hold.
Commutative Properties
Associative Properties
Identity Properties There is a real number 0 such that
and
There is a real number 1 such that
and
Inverse Properties For each real number a, there is a single real
number such that
and
For each nonzero real number a, there is a single
real number such that
and
Distributive Properties a 1 bc 2 abac 1 bc 2 abaca
1
a
a#^1 #a1.
a
1
1
aa 1 a 2 0 1 a 2 a0.
- a
a# 1 a 1 #aa.
a 0 a 0 aa.
1 ab 2 ca 1 bc 2
1 ab 2 ca 1 bc 2
abba abba
NOW TRY
EXERCISE 10
Write each expression with-
out parentheses.
(a)
(b) - 12 x- 5 y- 72
- 12 - r (^2) Using the Distributive Property to Remove (Clear) Parentheses
Write each expression without parentheses.
(a)
Distributive property=- 2 y- 3 Multiply.
=- 1 # 2 y+ 1 - 12 # 3
= - 1 # 12 y+ 32 - a=- 1 #a
- 12 y+ 32
EXAMPLE 10
The symbol
indicates a
factor of - 1.(b)
= 9 w+ 2
= - 11 - 9 w 2 - 11 - 22
= - 11 - 9 w - 22
- 1 - 9 w- 22 We can also interpret the negative
sign in front of the parentheses to
mean the oppositeof each of the
terms within the parentheses.(c)
= x+ 3 y- 6 z -1(- 1 x)= 1 x=x NOW TRY
=- 11 - 1 x 2 - 11 - 3 y 2 - 116 z 2
= - 11 - 1 x- 3 y+ 6 z 2
- 1 - x- 3 y+ 6 z 2
NOW TRY ANSWERS
- (a)
(b)- 2 x+ 5 y+ 7- 2 +r
Be careful with signs.The symbol may be interpreted as Using this result and the distrib-
utive property, we can remove (or clear) parentheses from some expressions.
- a - 1 #a.
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