Properties of Real Numbers

Example 5 Proof of a Property of Negation
Prove that
(-1)a = -a
(You may use any of the properties of equality and properties of zero .)

Solution
At first glance, it is a little difficult to see what you are being asked to prove.
However, a good way to start is to consider carefully the definitions of each of the
three numbers in the equation.
a = given real number
-1 = the additive inverse of 1
-a = the additive inverse of a

By showing that (-1)a has the same properties as the additive inverse of a, you
will be showing that (-1)a must be the additive inverse of a.

Multiplicative Identity Property
Distributive Property
Additive Inverse Property
Multiplication Property of Zero

Because you have shown that you can now use the fact that
-a + a = 0 to conclude that . From this, you can
complete the proof as follows.

Shown in first part of proof
Cancellation Property of Addition

The list of additional properties of real numbers forms a very important part
of algebra . Knowing the names of the properties is not especially important, but
knowing how to use each property is extremely important. The next two examples
show how several of the properties are used to solve common problems in algebra.

Example 6 Applying Properties of Real Numbers
In the solution of the equation b + 2 = 6, identify the property of real numbers
that justifies each step .

Solution
b + 2 = 6Original equation

Solution Step	Property
	Addition Property of Equality Associative Property of Addition Additive Inverse Property Additive Identity Property

Example 7 Applying the Properties of Real Numbers
In the solution of the equation 3a = 9, identify the property of real numbers that
justifies each step.

3a = 9  Original equation

Solution Step	Property
	Multiplication Property of Equality Associative Property of Multiplication Multiplicative Inverse Property Multiplicative Identity Property

P.2 Exercises

In Exercises 1–28, name the property of real numbers that
justifies the statement.

In Exercises 29–38, use the property of real numbers to fill
in the missing part of the statement.
29. Associative Property of Multiplication

30. Commutative Property of Addition

31. Commutative Property of Multiplication

32. Associative Property of Addition

33. Distributive Property

34. Distributive Property

35. Commutative Property of Addition

36. Additive Inverse Property

37. Multiplicative Identity Property

38. Additive Identity Property

In Exercises 39–46, give (a) the additive inverse and (b) the
multiplicative inverse of the quantity.

In Exercises 47–54, rewrite the expression using the
Associative Property of Addition or the Associative Property
of Multiplication.

In Exercises 55–62, rewrite the expression using the
Distributive Property.

In Exercises 63–68, the right side of the equation is not
equal to the left side. Change the right side so that it is
equal to the left side.

In Exercises 69 and 70, use the properties of real
numbers to prove the statement.
69. If ac = bc and c≠0, then a = b
70. (-1)(-a) = a

In Exercises 71–74, identify the property of real numbers
that justifies each step.

In Exercises 75–80, use the Distributive Property to perform
the arithmetic mentally . For example, you work in an industry
where the wage is $14 per hour with “time and a half ”
for overtime. So, your hourly wage for overtime is

Dividends In Exercises 81–84, the dividends paid per
share of common stock by the Proctor & Gamble Company
for the years 1994 through 2001 are approximated by the
expression
0.113t + 0.13.

In this expression, the dividend per share is measured in
dollars and t represents the year, with corresponding
to 1994 (see figure). (Source: Proctor & Gamble
Company)

81. Use the graph to approximate the dividend paid in
1999.
82. Use the expression to approximate the annual
increase in the dividend paid per share.
83. Use the expression to forecast the dividend per share
in 2004.
84. In 2000, the actual dividend paid per share of
common stock was $1.28. Compare this with the
approximation given by the expression.
85. Geometry The figure shows two adjoining rectangles.
Find the total area of the rectangles in two
ways.

86. Geometry The figure shows two adjoining rectangles.
Find the total area of the two rectangles in two
ways.

Synthesis
True or False? In Exercises 87–90, de termine whether
the statement is true or false. Justify your answer.

91. Does every real number have a multiplicative
inverse? Explain.
92. What is the additive inverse of a real number? Give
an example of the Additive Inverse Property.
93. What is the multiplicative inverse of a real number?
Give an example of the Multiplicative Inverse
Property.
94. State the Multiplication Property of Zero.
95. Explain how the Addition Property of Equality can
be used to al low you to subtract the same number
from each side of an equation.
96. Investigation You define a new mathematical
operation using the symbol . This operation is
defined as
(a) Is this operation commutative? Explain.
(b) Is this operation associative? Explain.