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Section 1.7
Multiplication
and Division
of Real
Numbers
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Objective 1
Multiply real numbers.
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The Product of Two Real Numbers
• The product of two real numbers with different
signs is found by multiplying their absolute
values. The product is negative.
• The product of two real numbers with same
signs is found by multiplying their absolute
values. The product is positive.
• The product of 0 and any real number is 0.
Thus, for any real number a,
a  0  0 and 0  a  0.
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Multiplying Real Numbers
9(5)  45
Same sign: positive product
2( 3)  6
Same sign: positive product
7(4)  28
Different signs: negative product
0(7)  0
Product of 0 and any real number 0
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Objective 1: Example
1 4
1a. Multiply:  
3 7
1 4
1 4
  
3 7
37
4

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Objective 1: Example
1b. Multiply: (  12)(  3)
(  12)(  3)  36
1c. Multiply: (  543)(0)
(  543)(0)  0
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Objective 2
Multiply more than two real numbers.
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Multiplying
Multiplying More Than Two Numbers
1. Assuming that no factor is zero.
• The product of an even number of negative
numbers is positive.
• The product of an odd number of negative
numbers is negative.
The multiplication is performed by multiplying
the absolute values of the given numbers.
2. If any factor is 0, the product is zero.
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Example
Multiply: (  3)(  5)(2)( 7)
(  3)(  5)(2)( 7)  210
Multiply
absolute
values:
3  5  2  7  210
Odd number of negative
numbers: negative
product.
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Example
Multiply: (  2)(  4)(6)( 1)( 3)
(  2)(  4)(6)( 1)( 3)  144
Multiply
absolute
values:
2  4  6  1 3  144
Even number of
negative numbers:
positive product.
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Objective 2: Example
2a.
Multiply: (  2)(3)(  1)(4)
(  2)(3)( 1)(4)  24
2b.
Multiply: (  1)(  3)(2)(  1)(5)
(  1)(  3)(2)(  1)(5)  30
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Objective 3
Find multiplicative inverses.
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Objective 3: Example
3a.
Find the multiplicative inverse of 7.
1
The multiplicative inverse of 7 is because
7
1
7   1.
7
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Objective 3: Example
3b.
Find the multiplicative inverse of 
7
.
13
13
7
The multiplicative inverse of 
is 
13
7
 7  13 
because       1.
 13  7 
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Objective 4
Use the definition of division.
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Definition of Division
If a and b are real numbers and b is not 0,
then the quotient of a and b is defined as
1
ab  a
b
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Objective 4: Example
4a.
Use the definition of division to find the
quotient: 28  7
1
28  7  28 
7
 4
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Objective 4: Example
4b.
Use the definition of division to find the
16
quotient:
2
16
 1
 16    
2
 2
8
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Objective 5
Divide real numbers.
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Quotient of Two Real Numbers
• The quotient of two real numbers with different
signs is found by dividing their absolute values.
The quotient is negative.
• The quotient of two real numbers with the same
sign is found by dividing their absolute values.
The quotient is positive.
• Division of a nonzero number by zero is
undefined.
• Any nonzero number divided into 0 is 0.
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Example
16
Divide:
4
16
 4
4
12
Divide:
2
12
6
2
Different signs:
Negative
Quotient
Same signs:
Positive
Quotient
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Objective 5: Example
5a.
2 5
Divide:  
3 4
2 5
2 4
   
3 4
3 5
8

15
5b.
0
Divide:
5
0
0
5
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Objective 6
Simplify algebraic expressions involving
multiplication.
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Additional Properties of Multiplication
• Identity Property of Multiplication
a 1  a
1 a  a
• Inverse Property of Multiplication If a is
not 0:
1
a 1
a
1
a 1
a
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Additional Properties of Multiplication
• Multiplication Property of 1
1 a  a
a( 1)  a
• Double Negative Property
The opposite of a is a.
( a )  a
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Negative Signs and Parentheses
• If a negative sign precedes parentheses,
remove the parentheses and change the sign of
every term within the parentheses.
• Examples:
(8 x  7)  8 x  ( 7)
( 6 x  5)  6 x  5
( 5 x  3)  5 x  3
Note: By doing this, we are using the Distributive
Law and distributing the factor of 1.
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Objective 6: Example
6a.
Simplify: 7(3 x  4)
7(3 x  4)  7(3 x )  7( 4)
 21x  28
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Example
Simplify: 3(2 x  5)  (4 x  7)
 3  2x  3  5  4 x  7
 6 x  15  4 x  7
 (6 x  4 x )  ( 15  7)
 2 x  22
 2 x  22
Distributive property.
Multiply.
Group like terms.
Combine like terms.
Express addition of an
additive inverse as
subtraction.
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Objective 6: Example
6b.
Simplify: 4(3 y  7)  (13 y  2)
4(3 y  7)  (13 y  2)  12 y  28  13 y  2
 12y  13 y  28  2
 1y  26
  y  26
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Objective 7
Determine whether a number is a
solution of an equation.
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Objective 7: Example
7.
Determine whether 3 is a solution of
2 x  5  8 x  7.
2x  5  8 x  7
2( 3)  5  8( 3)  7
6  5  24  7
11  17, false
3 is not a solution of the equation.
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Objective 8
Use mathematical models involving
multiplication and division.
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Objective 8: Example
8. The data for doctorate degrees earned by men
can be described by M  0.6n  64.4, where M
is the percentage of doctorate degrees
awarded to men n years after 1989. According
to this mathematical model what percentage of
doctorate degrees are projected to be received
by men in 2014?
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Objective 8: Example (cont)
M  0.6n  64.4
M  0.6(25)  64.4
 15  64.4
 49.4
According to this model, 49.4% of doctorate
degrees will be awarded to men in 2014.
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