Properties of
Real Numbers
Objective
Identify and use
properties of real
numbers.
Why learn this?
You can use properties of real
numbers to quickly calculate tips in
your head. (See Example 3.)
Andrew Toos/CartoonResource.com
1-2
The four basic math operations are
addition, subtraction, multiplication,
and division. Because subtraction is
addition of the opposite and division
is multiplication by the reciprocal,
the properties of real numbers
focus on addition and multiplication.
Properties of Real Numbers
Identities and Inverses
For all real numbers n,
WORDS
NUMBERS
ALGEBRA
3+0=3
n+0=0+n=n
Additive Identity Property
The sum of a number and 0, the additive
identity, is the original number.
Multiplicative Identity Property
The product of a number and 1, the
multiplicative identity, is the original
number.
2
2 ·1=_
_
3
3
n·1=1·n=n
Additive Inverse Property
The sum of a number and its opposite,
or additive inverse, is 0.
5 + (-5) = 0
n + (-n) = 0
_
n · 1 = 1 (n ≠ 0)
n
Multiplicative Inverse Property
The product of a nonzero number and its
reciprocal, or multiplicative inverse, is 1.
8· 1 =1
8
_
Recall from previous courses that the opposite of any number a is -a and
the reciprocal of any nonzero number a is __a1 .
EXAMPLE
1
Finding Inverses
Find the additive and multiplicative inverse of each number.
A -9
additive inverse: 9
Check -9 + 9 = 0 ✔
1
multiplicative inverse: _
-9
Check -9 · 1 = 1 ✔
-9
(_ )
14
The opposite of -9 is -(-9) = 9.
The Additive Inverse Property holds.
1 .
The reciprocal of -9 is _
-9
The Multiplicative Inverse Property holds.
Chapter 1 Foundations for Functions
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Find the additive and multiplicative inverse of each number.
B
_4
5
4
additive inverse: -_
5
5
multiplicative inverse: _
4
4 is _
The opposite of _
-4.
5
5
5.
4 is _
The reciprocal of _
5
4
Find the additive and multiplicative inverse of each number.
1b. -0.01
1a. 500
Properties of Real Numbers
Addition and Multiplication
For all real numbers a and b,
WORDS
NUMBERS
ALGEBRA
2+3=5
2(3) = 6
a+b∈
ab ∈ 7 + 11 = 11 + 7
7(11) = 11(7)
a+b=b+a
ab = ba
(5 + 3) + 7 =
5 + (3 + 7)
(5 · 3)7 = 5(3 · 7)
(a + b) + c =
a + (b + c)
(ab)c = a(bc)
5(2 + 8) = 5(2) + 5(8)
(2 + 8)5 = (2)5 + (8)5
a(b + c) = ab + ac
(b + c)a = ba + ca
Closure Property
The sum or product of any two
real numbers is a real number.
Commutative Property
You can add or multiply real
numbers in any order without
changing the result.
Based on the Closure
Property, the real
numbers are said
to be closed under
addition and closed
under multiplication.
Associative Property
The sum or product of three
or more real numbers is the
same regardless of the way the
numbers are grouped.
Distributive Property
When you multiply a sum by a
number, the result is the same
whether you add and then
multiply or whether you multiply
each term by the number and
then add the products.
EXAMPLE
2
Identifying Properties of Real Numbers
Identify the property demonstrated by each equation.
A
(3 √3 + 5)2 = (3 √3)2 + (5)2
Distributive Property
B (3 + 6) + (-6) = 3 + [6 + (-6)]
The 2 has been distributed to
each term.
The numbers have been regrouped.
Associative Property of Addition
Identify the property demonstrated by each equation.
2a. 9 √
2 = ( √
2 )9
2b. 9(12π) = (9 · 12)π
You can apply the properties of real numbers to simplify numeric expressions
and solve problems mentally.
1- 2 Properties of Real Numbers
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EXAMPLE
3
Tea
Consumer Economics Application
Use mental math to find a 15% tip for the bill shown.
Think: 15% = 10% + 5%
Tax
(10% + 5% )24.80
1.40
$24.80
Distributive
Property
10% (24.80) + 5% (24.80)
3.20
Subtotal 23.4
0
Think: Find 10% of $24.80
10% (24.80) = 2.480 = 2.48
1 (10%)
Think: 5% = _
2
1 (2.48 ) = 1.24
_
2
2.48 + 1.24 = 3.72
Move the decimal point left 1 place.
5% is half of 10% so find half of 2.48.
Add 10% of 24.80 to 5% of 24.80.
A 15% tip for a meal that totaled $24.80 is $3.72.
3. Use mental math to find a 20% discount on a $15.60 shirt.
EXAMPLE
4
Classifying Statements as Sometimes, Always, or Never True
Classify each statement as sometimes, always, or never true. Give
examples or properties to support your answer.
A c + d = c when d = 2
By the Additive Identity Property,
c + 0 = c , so c + d = c is only true
when d = 0, not when d = 2.
never true
counterexample: 1 + 2 ≠ 1
B a-c=c-a
sometimes true
true example: 5 - 5 = 5 - 5
false example: 5 - 2 ≠ 2 - 5
True and false examples exist. The
statement is true when a = c and
false when a ≠ c.
Classify each statement as sometimes, always, or never true.
Give examples or properties to support your answer.
4a. a + (-a) = b + (-b)
4b. a - (b + c) = (a - b) + (a - c)
THINK AND DISCUSS
1. Explain whether the Commutative Property applies to subtraction
and division.
2. Tell why zero has no
multiplicative inverse.
3. GET ORGANIZED Copy
and complete the graphic
organizer. In each box,
write an example of the
property indicated.
16
Property
Addition
Multiplication
Identity
Inverse
Associative
Commutative
Distributive
Chapter 1 Foundations for Functions
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