Ch. 1.6 power point
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Chapter 1
Section 6
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
1.6
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Multiplying and Dividing Real Numbers
Find the product of a positive number and a negative
number.
Find the product of two negative numbers.
Identify factors of integers.
Use the reciprocal of a number to apply the definition of
division.
Use the rules for order of operations when multiplying
and dividing signed numbers.
Evaluate expressions involving variables.
Interpret words and phrases involving multiplication and
division.
Translate simple sentences into equations.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Multiplying and Dividing Real Numbers
The result of multiplication is called the product. We already
know how to multiply positive numbers, and we know that the
product of two positive numbers is positive.
We also know that the product of 0 and any positive number is
0, so we extend that property to all real numbers.
Multiplication by Zero says,
for any real number x,
x0 0 .
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Objective 1
Find the product of a positive and
negative number.
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Find the product of a positive number
and a negative number.
The product of a 3(−1) represents the sum
1 (1) (1) 3.
Also,
and
3(2) 2 (2) (2) 6
3(3) 3 (3) (3) 9 .
These results maintain the pattern, which suggests the rule for
Multiplying Numbers with Different Signs;
For any positive real numbers x and y,
x( y ) ( xy ) and ( x) y ( xy ) .
That is, the product of two numbers with opposite signs is
negative.
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EXAMPLE 1
Multiplying a Positive and a
Negative Number
Find the product.
Solution:
80
85 2
5
5
16
32
822
2
32
4.56 2 9.12
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Objective 2
Find the product of two negative
numbers.
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Find the product of two negative
numbers.
The rule for Multiplying Two Negative Numbers states
that:
For any positive real numbers x and y,
x( y ) xy
That is, the product of two negative numbers is positive.
Example: 5(4) 20
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EXAMPLE 2
Multiplying Two Negative
Numbers
Find the product.
Solution:
3 2
6
3 2
4 5
20
10 2
3
10
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Objective 3
Identify factors of integers.
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Identify factors of integers.
In Section 1.1, the definition of a factor was given for whole
numbers. The definition can be extended to integers. If the
product of two integers is a third integer, then each of the two
integers is a factor of the third.
The table below show several examples of integers and
factors of those integers.
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Objective 4
Use the reciprocal of a number to
apply the definition of division.
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Use the reciprocal of a number to apply
the definition of division.
The quotient of two numbers is found by multiplying by the
reciprocal, or multiplicative inverse. By definition, since
1 8
5 4 20
8 1 and
1 ,
8 8
4 5 20
1
5
4
the reciprocal or multiplicative inverse of 8 is and of is .
8
4
5
Pairs of numbers whose product is 1 are called reciprocals, or
multiplicative inverses, of each other.
Suppose that k is to be the multiplicative inverse of 0. Then
k · 0 should equal 1. But, k · 0 = 0 for any real number. Since
there is no value of k that is a solution of the equation k · 0 = 1,
the following statement can be made:
0 has no multiplicative inverse
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Use the reciprocal of a number to apply
the definition of division. (cont’d)
The Definition of Division says that,
for any real numbers x and y, with y ≠ 0,
x
1
x
y
y
That is, to divide two numbers, multiply the first by the
reciprocal, or multiplicative inverse, of the second.
If a division problem involves division by 0, write
“undefined.”
1
In the expression
, x cannot have the value of 2 because then
x2
the denominator would equal 0 and the fraction would be undefined.
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EXAMPLE 3
Using the Definition of Division
Find each quotient, using the definition of division.
Solution:
36
1
36 6
6
6
12.56
10
12.56 31.4
0.4
4
2 25
50
25
10 24 10 5
168
2 84
84
7 24
7 5
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Dividing Signed Numbers
When dividing fractions, multiplying by the reciprocal works
well. However, using the definition of division directly with
integers is awkward.
It is easier to divide in the usual way and then determine the
sign of the answer.
The quotient of two numbers having the same sign is positive.
The quotient of two numbers having different signs is negative.
Examples:
15
3 ,
5
15
15
3
3, and
5
5
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EXAMPLE 4
Dividing Signed Numbers
Find each quotient.
Solution:
16
1
16 8
2
2
16.4
1
16.4
8
2.05
2.05
3
1 2
1 3
8
4 3
4 2
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Objective 5
Use the rules for order of
operations when multiplying and
dividing signed numbers.
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EXAMPLE 5
Using the Rules for Order of
Operations
Perform each indicated operation.
3(4) 2(6)
Solution:
12 12 0
21
3
6(8) (3)9 48 27
14
2
2 7
(2) 4 (3)
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Objective 6
Evaluate expressions involving
variables.
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Evaluating Expressions for
Numerical Values
EXAMPLE 6
Evaluate 2 x 2 4 y 2 if x 2 and y 3.
Solution:
2(2) 4(3)
2
2
2(4) 4(9)
8 36
28
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Objective 7
Interpret words and phrases
involving multiplication and
division.
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Interpret words and phrases involving
multiplication.
The word product refers to multiplication. The table gives
other key words and phrases that indicate multiplication in
problem solving.
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EXAMPLE 7
Interpreting Words and Phrases
Involving Multiplication
Write a numerical expression for the phrase and
simplify the expression.
Three times the difference between 4 and −11.
Solution: 3 4 (11) 3 15 45
Three-fifths of the sum of 2 and −7.
3
3
2 (7) 5 3
5
5
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Interpret words and phrases involving
division.
The word quotient refers to division. In algebra, quotients
are usually represented with a fraction bar; the symbol ÷ is
seldom used. The table gives some key phrases associated with
division.
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EXAMPLE 8
Interpreting Words and Phrases
Involving Division
Write a numerical expression for the phrase and
simplify the expression.
The product of −9 and 2, divided by the
difference between 5 and −1.
Solution:
18
9(2)
3
6
5 ( 1)
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Objective 8
Translate simple sentences into
equations.
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EXAMPLE 9
Translating Sentences into
Equations
Write the sentence in symbols, using x as the variable.
Then find the solution from the list of integers between
−12 and 12, inclusive.
The quotient of a number and −2 is 6.
x
6
Solution:
2
Here, x must be a negative number since the
denominator is negative and the quotient is
12
6 , the solution is −12.
positive. Since
2
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