SECTION 1.4 Properties of Real Numbers 33
(d)
Definition of subtraction
Distributive property(e)
Because there is no common number or variable here, we cannot use the distrib-
utive property to rewrite the expression.
(f )
NOW TRYThe distributive property can also be used for subtraction (Example 1(d)),so
OBJECTIVE 2 Use the identity properties.The number 0 is the only number
that can be added to any number to get that number, leaving the identity of the number
unchanged. For this reason, 0 is called the identity element for addition,or the
additive identity.
In a similar way, multiplying any number by 1 leaves the identity of the number
unchanged. Thus, 1 is the identity element for multiplication, or the multiplicative
identity.
The identity propertiessummarize this discussion and extend these properties
from arithmetic to algebra.
a 1 bc 2 abac.
= 6 x+ 12 y- 18 z
= 6 x+ 612 y 2 + 61 - 3 z 2
61 x+ 2 y- 3 z 2
5 p+ 7 q
=- 4 r
= 33 + 1 - 724 r
= 3 r+ 1 - 7 r 2
3 r - 7 r
Identity PropertiesFor any real number a, the following are true.
a# 1 1 #aa
a 0 0 aa
The identity properties leave the identity of a real number unchanged.Think of
a child wearing a costume on Halloween. The child’s appearance is changed, but his
or her identity is unchanged.
Using the Identity PropertySimplify each expression.
(a)
Identity property; or 1m
Distributive property
Add inside parentheses.(b)
Identity property
Distributive property= 2 y Add inside parentheses.
= 11 + 12 y
= 1 y+ 1 y
y+y
= 13 m
= 112 + 12 m
= 12 m+ 1 m m= 1 #m,
12 m+m
EXAMPLE 2 1 #aa
NOW TRY
EXERCISE 1
Use the distributive property
to rewrite each expression.
(a)
(b) 4 k- 12 k
- 213 x-y 2
NOW TRY ANSWERS
- (a)- 6 x+ 2 y (b) - 8 k